fourier-mukai and nahm transforms in geometry and mathematical physics

Joseph Oesterle

Progress in Mathematics

Hyman Bass

Alan Weinstein

Volume 276

Series Editors

BirkhauserBoston • Basel • Berlin

Fourier–Mukai and NahmTransforms in Geometry and Mathematical Physics

Claudio BartocciUgo BruzzoDaniel Hernández Ruipérez

ISBN 978-0-8176-3246-5 e-ISBN 978-0-8176–4663-9

Claudio Bartocci

Università di Genova

Ugo BruzzoScuola Internazionale Superiore di

Departamento de Matemáticas

Universidad de Salamanca

Cover Design by Joseph Sherman

Library of Congress Control Number: 2009926479

Printed on acid-free paper

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Dipartimento di Matematica

Genova, Italy

Salamanca, [email protected]

[email protected] Trieste, [email protected]

Birkhäuser Boston is part of Springer Science+Business Media (www.springer.com)

DOI 10.1007/b11801

+Business Media, LLC 2009© Birkhäuser Boston, a part of Springer Science

Springer Dordrecht Heidelberg London New York

Studi Avanzati and Istituto Nazionaledi Fisica Nucleare

and Instituto Universitario de Fisica Fundamental y Matemáticas

18E30, 19K56, 53C07, 58J20Mathematics Subject Classification (2000): 14-02, 14D21, 14D20, 14E05, 14F05, 14J28, 14J32, 14J81, 14K05,

Daniel Hernández Ruipérez

Contents

Preface xi

Acknowledgments xv

1 Integral functors 1

1.1 Notation and preliminary results . . . . . . . . . . . . . . . . . . . 2

1.2 First properties of integral functors . . . . . . . . . . . . . . . . . . 5

1.2.1 Base change formulas . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Fully faithful integral functors . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Strongly simple objects . . . . . . . . . . . . . . . . . . . . 19

1.4 The equivariant case . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.1 Equivariant and linearized derived categories . . . . . . . . 24

1.4.2 Equivariant integral functors . . . . . . . . . . . . . . . . . 29

1.5 Notes and further reading . . . . . . . . . . . . . . . . . . . . . . . 30

2 Fourier-Mukai functors 31

2.1 Spanning classes and equivalences . . . . . . . . . . . . . . . . . . . 32

2.1.1 Ample sequences . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.2 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Orlov’s representability theorem . . . . . . . . . . . . . . . . . . . 44

2.2.1 Resolution of the diagonal . . . . . . . . . . . . . . . . . . . 44

2.2.2 Uniqueness of the kernel . . . . . . . . . . . . . . . . . . . . 51

2.2.3 Existence of the kernel . . . . . . . . . . . . . . . . . . . . . 54

vi Contents

2.3 Fourier-Mukai functors . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3.1 Some geometric applications of Fourier-Mukai functors . . . 61

2.3.2 Characterization of Fourier-Mukai functors . . . . . . . . . 71

2.3.3 Fourier-Mukai functors between moduli spaces . . . . . . . 76


3 Fourier-Mukai on Abelian varieties 81

3.1 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.2 The transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3 Homogeneous bundles . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4 Fourier-Mukai transform and the geometry of Abelian varieties . . 91

3.4.1 Line bundles and homomorphisms of Abelian varieties . . . 91

3.4.2 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4.3 Picard sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.5 Some applications of the Abelian Fourier-Mukai transform . . . . . 97

3.5.1 Moduli of semistable sheaves on elliptic curves . . . . . . . 97

3.5.2 Preservation of stability for Abelian surfaces . . . . . . . . 102

3.5.3 Symplectic morphisms of moduli spaces . . . . . . . . . . . 104

3.5.4 Embeddings of moduli spaces . . . . . . . . . . . . . . . . . 106


4 Fourier-Mukai on K3 surfaces 111

4.1 K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2 Moduli spaces of sheaves and integral functors . . . . . . . . . . . 116

4.3 Examples of transforms . . . . . . . . . . . . . . . . . . . . . . . . 122

4.3.1 Reflexive K3 surfaces . . . . . . . . . . . . . . . . . . . . . . 124

4.3.2 Duality for reflexive K3 surfaces . . . . . . . . . . . . . . . 125

4.3.3 Homogeneous bundles . . . . . . . . . . . . . . . . . . . . . 131

4.3.4 Other Fourier-Mukai transforms on K3 surfaces . . . . . . . 133

4.4 Preservation of stability . . . . . . . . . . . . . . . . . . . . . . . . 139

4.5 Hilbert schemes of points on reflexive K3 surfaces . . . . . . . . . . 142


5 Nahm transforms 147

5.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Contents vii

5.1.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.1.2 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.1.3 The Hitchin-Kobayashi correspondence . . . . . . . . . . . 153

5.1.4 Dirac operators and index bundles . . . . . . . . . . . . . . 155

5.2 The Nahm transform for instantons . . . . . . . . . . . . . . . . . . 158

5.2.1 Definition of the Nahm transform . . . . . . . . . . . . . . . 158

5.2.2 The topology of the transformed bundle . . . . . . . . . . . 161

5.2.3 Line bundles on complex tori . . . . . . . . . . . . . . . . . 161

5.2.4 Nahm transform on flat 4-tori . . . . . . . . . . . . . . . . . 164

5.3 Compatibility between Nahm and Fourier-Mukai . . . . . . . . . . 165

5.3.1 Relative differential operators . . . . . . . . . . . . . . . . . 165

5.3.2 Relative Dolbeault complex . . . . . . . . . . . . . . . . . . 166

5.3.3 Relative Dirac operators . . . . . . . . . . . . . . . . . . . . 170

5.3.4 Kahler Nahm transforms . . . . . . . . . . . . . . . . . . . 171

5.4 Nahm transforms on hyperkahler manifolds . . . . . . . . . . . . . 173

5.4.1 Hyperkahler manifolds . . . . . . . . . . . . . . . . . . . . . 173

5.4.2 A generalized Atiyah-Ward correspondence . . . . . . . . . 174

5.4.3 Fourier-Mukai transform of quaternionic instantons . . . . . 178

5.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180


6 Relative Fourier-Mukai functors 183

6.1 Relative integral functors . . . . . . . . . . . . . . . . . . . . . . . 184

6.1.1 Base change formulas . . . . . . . . . . . . . . . . . . . . . 185

6.1.2 Fourier-Mukai transforms on Abelian schemes . . . . . . . . 188

6.2 Weierstraß fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.2.1 Todd classes . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.2.2 Torsion-free rank one sheaves on elliptic curves . . . . . . . 192

6.2.3 Relative integral functors for Weierstraß fibrations . . . . . 193

6.2.4 The compactified relative Jacobian . . . . . . . . . . . . . . 197

6.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.2.6 Topological invariants . . . . . . . . . . . . . . . . . . . . . 201

6.3 Relatively minimal elliptic surfaces . . . . . . . . . . . . . . . . . . 204

6.4 Relative moduli spaces for Weierstraß elliptic fibrations . . . . . . 208

viii Contents

6.4.1 Semistable sheaves on integral genus one curves . . . . . . . 208

6.4.2 Characterization of relative moduli spaces . . . . . . . . . 213

6.5 Spectral covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.6 Absolutely stable sheaves on Weierstraß fibrations . . . . . . . . . 220

6.6.1 Preservation of absolute stability for elliptic surfaces . . . . 221

6.6.2 Characterization of moduli spaces on elliptic surfaces . . . . 225

6.6.3 Elliptic Calabi-Yau threefolds . . . . . . . . . . . . . . . . . 228


7 Fourier-Mukai partners and birational geometry 233

7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.2 Integral functors for quotient varieties . . . . . . . . . . . . . . . . 238

7.3 Fourier-Mukai partners of algebraic curves . . . . . . . . . . . . . . 242

7.4 Fourier-Mukai partners of algebraic surfaces . . . . . . . . . . . . 242

7.4.1 Surfaces of Kodaira dimension 2 . . . . . . . . . . . . . . . 245

7.4.2 Surfaces of Kodaira dimension −∞ that are not elliptic . . 245

7.4.3 Relatively minimal elliptic surfaces . . . . . . . . . . . . . . 248

7.4.4 K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.4.5 Abelian surfaces . . . . . . . . . . . . . . . . . . . . . . . . 253

7.4.6 Enriques surfaces . . . . . . . . . . . . . . . . . . . . . . . . 254

7.4.7 Nonminimal projective surfaces . . . . . . . . . . . . . . . . 256

7.5 Derived categories and birational geometry . . . . . . . . . . . . . 257

7.5.1 A removable singularity theorem . . . . . . . . . . . . . . . 258

7.5.2 Perverse sheaves . . . . . . . . . . . . . . . . . . . . . . . . 264

7.5.3 Flops and derived equivalences . . . . . . . . . . . . . . . . 272

7.6 McKay correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 275

7.6.1 An equivariant removable singularity theorem . . . . . . . . 276

7.6.2 The derived McKay correspondence . . . . . . . . . . . . . 277


A Derived and triangulated categories 281

A.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

A.2 Additive and Abelian categories . . . . . . . . . . . . . . . . . . . . 283

A.3 Categories of complexes . . . . . . . . . . . . . . . . . . . . . . . . 287

Contents ix

A.3.1 Double complexes . . . . . . . . . . . . . . . . . . . . . . . 292

A.4 Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

A.4.1 The derived category of an Abelian category . . . . . . . . 295

A.4.2 Other derived categories . . . . . . . . . . . . . . . . . . . . 300

A.4.3 Triangles and triangulated categories . . . . . . . . . . . . . 303

A.4.4 Differential graded categories . . . . . . . . . . . . . . . . . 307

A.4.5 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . 312

A.4.6 Some remarkable formulas in derived categories . . . . . . . 328

A.4.7 Support and homological dimension . . . . . . . . . . . . . 335

B Lattices 339

B.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

B.2 The discriminant group . . . . . . . . . . . . . . . . . . . . . . . . 341

B.3 Primitive embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 342

C Miscellaneous results 347

C.1 Relative duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

C.2 Pure sheaves and Simpson stability . . . . . . . . . . . . . . . . . . 351

C.3 Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

D Stability conditions for derived categories 359

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

D.2 Bridgeland’s stability conditions . . . . . . . . . . . . . . . . . . . 362

D.2.1 Definition and Bridgeland’s theorem . . . . . . . . . . . . . 363

D.2.2 An example: stability conditions on curves . . . . . . . . . . 369

D.2.3 Bridgeland’s deformation lemma . . . . . . . . . . . . . . . 371

D.3 Stability conditions on K3 surfaces . . . . . . . . . . . . . . . . . . 373

D.3.1 Bridgeland’s theorem . . . . . . . . . . . . . . . . . . . . . . 374

D.3.2 Construction of stability conditions . . . . . . . . . . . . . . 375

D.3.3 The covering map property . . . . . . . . . . . . . . . . . . 380

D.3.4 Wall and chamber structure . . . . . . . . . . . . . . . . . . 382

D.3.5 Sketch of the proof of Theorem D.19 . . . . . . . . . . . . . 383

D.4 Moduli stacks and invariants of semistable objects on K3 surfaces . 385

D.4.1 Moduli stack of semistable objects . . . . . . . . . . . . . . 385

D.4.2 Sketch of the proof of Theorem D.35 . . . . . . . . . . . . . 386

x Contents

D.4.3 Counting invariants and Joyce’s conjecture for K3 surfaces 391

D.4.4 Some ideas from the proof of Theorem D.45 . . . . . . . . . 392

References 397

Subject index 419

Preface

A fundamental question in geometry is to find invariants for a given class of geo-metric objects. In the case of algebraic varieties, natural invariants are the Chowgroups and the algebraic K-theory. In looking for finer invariants one could thinkof the category of coherent sheaves; however, it is known that two projective va-rieties are isomorphic if and only if the respective categories of coherent sheavesare equivalent [120]. A straightforward extension of this idea is to look at thederived category of coherent sheaves. That notion was used by Grothendieck andVerdier as an appropriate framework for their theory of duality; moreover, in thelate 1970s Beılinson gave a simple characterization of the derived categories ofprojective spaces. Derived categories of coherent sheaves appeared again in thefundamental paper by Shigeru Mukai [224], where the integral functor now called“Fourier-Mukai transform” was introduced.

A radical change of perspective in connection with derived categories tookplace with a result of Bondal and Orlov, according to which the derived categoryof a projective variety, whose canonical or anticanonical bundle is ample, fullydetermines the variety. Subsequent work by Orlov showed that any equivalence ofderived categories of coherent sheaves on two projective varieties X and Y is anintegral functor — i.e., a kind of “correspondence” induced by an object in thederived category of the product X × Y . This prompts us to consider the moregeneral question: to what extent does the derived category of coherent sheavesdetermine the underlying algebraic variety? And also, what is the relationshipbetween the group of automorphisms of the derived category of a projective varietyand the group of isomorphisms of the variety?

In the second half of the 1990s, physicists working in string theory got in-terested in triangulated categories. Actually, the quantum theories correspondingto some string models admit solitonic states (branes), which can be geometricallycharacterized as coherent sheaves supported on subvarieties of the compactifica-tion space of the string (in most cases a Calabi-Yau threefold). Thus, derivedcategories of coherent sheaves naturally come into play, and integral functors canbe exploited to describe some mirror dualities.

xii Preface

This book is about the study of integral functors. Given two projective vari-eties X and Y , an integral functor between the derived categories D(X) and D(Y )is a functor of the type

ΦK•

X→Y (E•) = RπY ∗(π∗XE•L⊗K•) ,

where K• is an object in D(X × Y ) (a complex of coherent sheaves on X × Y ,called the kernel of the integral functor) and πX , πY are the projections onto thetwo factors of X × Y (for precise definitions, the reader is referred to Chapter1). When an integral functor is an equivalence of categories, we shall call it aFourier-Mukai functor, and if in addition the kernel K• is concentrated (i.e., itreduces to a single coherent sheaf on X × Y ), the integral functor will be called aFourier-Mukai transform. The prototype of this kind of transform was defined byMukai in 1981: X is an Abelian variety, Y its dual variety, and the kernel is thePoincare bundle on the product X × Y .

Besides developing the basic theory of integral functors, we shall put specialemphasis on some of their applications to geometry and mathematical physics.As a matter of fact, one proves that whenever two projective varieties X and Y

have equivalent derived categories (i.e., they are Fourier-Mukai partners), thenY is a coarse moduli space of coherent sheaves on X, and vice versa. Thus, weshall be mostly concerned with applications to moduli spaces of sheaves. A firstexample is Mukai’s original transform, which establishes the equivalence betweenthe derived categories of an Abelian variety X and of its dual variety X, which isindeed the moduli space of flat line bundles on X. Other examples are provided bythe construction of Fourier-Mukai transforms for K3 surfaces, and by the relativeFourier-Mukai transforms. The latter play an important role in the so-called ho-mological mirror symmetry in string theory and in some constructions appearingin the theory of algebraically completely integrable systems.

We now give a cursory presentation of the contents of the book. Chapter 1 isdevoted to the foundations of the theory of integral functors. A key notion is that ofstrongly simple object, which provides a characterization of fully faithful integralfunctors (Theorem 1.27). This result — due to Bondal and Orlov [48] — will bea cornerstone on which we shall build many of the most fundamental theoremsproved in this book. In Chapter 2 we further develop the theory of these functors,considering the case when they are equivalences of derived categories. Orlov’srepresentability theorem [242] — whose proof involves the notion of spanningclasses for a derived category of coherent sheaves — shows that every equivalenceof derived categories is an integral functor. This explains the pervasiveness ofFourier-Mukai functors in the study of the geometry of algebraic varieties.

While the two initial chapters are framed in a rather general setting, inChapters 3 and 4 we study in some detail the cases of Abelian varieties and K3surfaces. In particular, in Chapter 3 we review Mukai’s original construction and

Preface xiii

present some applications. The first nontrivial instance of Fourier-Mukai transformwas obtained on K3 surfaces by the current authors [24]. In this case, X is a K3surface whose Neron-Severi group satisfies certain restrictions, Y is 2-dimensionalcompact component of the moduli space of stable bundles over X (which oneproves to be a K3 surface as well) and the kernel is the relevant universal sheaf.A remarkable feature of the Fourier-Mukai transform in the case of both Abelianand K3 surfaces is that, under suitable assumptions, it preserves the stability ofthe sheaves it operates on. Consequently, it supplies a helpful tool to investigatethe structure of moduli spaces of stables sheaves, as we show, e.g., in Section 4.5.

Chapter 5 is a digression in the realm of complex differential geometry. Ona compact Kahler manifold, Hermitian-Yang-Mills bundles and stable bundlesare related by the celebrated Hitchin-Kobayashi correspondence. Regarding anAbelian surface as a flat 4-dimensional real torus T and applying this correspon-dence, Mukai’s original transform translates into Nahm’s transform, introducedin the early 1980s by the physicist Werner Nahm to study periodic instantons onR4 [230]. This transform is based on an index-theoretic construction: thinking ofthe dual torus T as a space parameterizing a family of twisted Dirac operators,one can associate an index bundle to any instanton on T . After extending Nahm’stransform to a more general setting, we show how it relates with a Fourier-Mukaitransform. We examine in some detail the case when the base manifold carries ahyperkahler structure. Besides being interesting on its own, this perspective shedsnew light on some results obtained in the algebro-geometric framework: for in-stance, in Section 5.4, we provide a different proof of the preservation of stabilityfor bundles on Abelian and K3 surfaces.

In Chapter 6 we develop the machinery of relative Fourier-Mukai functors foralgebraic B-schemes. These are a particular kind of integral functors, whose kernelis an element of the derived category of the fibered product X ×B Y (here X andY are schemes over a base scheme B). We offer a quite comprehensive treatmentof elliptic fibrations X → B, dealing separately with the case when the fibrationadmits a Weierstraß model (allowing the base scheme to be of arbitrary dimen-sion) and the case of relatively minimal elliptic surfaces. In the first situation, westudy the moduli spaces of relatively semistable sheaves and discuss the notion ofspectral cover. If the total space X of the Weierstraß fibration has dimension 2or 3, the Fourier-Mukai transform establishes a correspondence between relativelysemistables sheaves on X and spectral data on the compactified relative Jacobianassociated with the fibration. When X is an elliptically fibered Calabi-Yau three-fold, this construction — originally obtained by Friedman, Morgan and Witten[113, 114] — is relevant to string theory.

The study of the Fourier-Mukai functor on elliptic surfaces is part of a muchwider research program, which we pursue in Chapter 7. Two projective varietiesare said to be Fourier-Mukai partners if there is an exact equivalence of trian-

xiv Preface

gulated categories between their bounded derived categories; in view of Orlov’srepresentability theorem, this amounts to the existence of a Fourier-Mukai functorbetween them. Section 7.4 is devoted to the classification of Fourier-Mukai part-ners of algebraic surfaces, a result that for minimal surfaces was first obtained byBridgeland and Maciocia. The problem of the birational invariance of the derivedcategory of a projective variety is dealt with in Section 7.5, while Bridgeland-King-Reid’s interpretation of the McKay correspondence is presented in Section7.6. The results presented in Chapter 7 are a clear indication of the significance ofFourier-Mukai transforms in algebraic geometry, as it has been recently pointedout by Kawamata and others.

We have made an effort to be self-contained, devoting three appendices topresent some preliminary material (respectively, derived categories, integral lat-tices and a miscellany of results in algebraic geometry), and proving most ofthe fundamental theorems from scratch. Generically, prerequisites for reading thisbook reduce to a basic knowledge of algebraic geometry at the level of Hartshorne[141] and, for Chapter 5, to some rudiments of differential geometry (manifolds,bundles, connections, and on a somehow more advanced level, some theory of el-liptic operators on spaces of sections of a vector bundle). In addition, we use quiteextensively some basic categorial language, and occasionally, we employ spectralsequences.

On the other hand, we did not aim at giving an exhaustive treatment of thesubject, which appears to be widely ramified and steadily growing. Among themost conspicuous omissions, we do not deal with the case of Fourier-Mukai func-tors on singular varieties and we leave out the important topic of autoequivalencesof derived categories. Furthermore, we mention only cursorily the recent develop-ments related to differential graded categories and derived algebraic geometry. Wehave tried to put a remedy to major and minor omissions adding a “notes andfurther reading” section at the end of each chapter. In this connection one shouldalso cite D. Huybrecht’s book on Fourier-Mukai transforms [153]. Should one tryto compare the two books in terms of their contents, one would notice that ourbook is more abundant in technical details, and somehow aims at a reasonably self-contained treatment of the arguments it touches. On the other hand, the choiceof topics in the two books is somehow different; in this sense, we believe that thetwo books nicely complement each other.

A final appendix serves as an introduction to one of the most interestingrecent developments related to integral functors, namely, the notion of stabilitycondition for derived categories.

Claudio BartocciUgo BruzzoDaniel Hernandez Ruiperez

April 2008

Acknowledgments

The project of writing this book saw the light in 2002 at a conference in Cookeville,Tennessee, organized by our friend Rafa l Ab lamowicz. It is very apt that he is thefirst person whom we thank. The project started with a fourth author, MarcosJardim, who at some stage preferred to withdraw. We owe many thanks to Marcos:without him this book would not have been written.

Appendix A, an introduction to derived categories that originates from aset of notes for a course at the School on Algebraic Geometry and Physics inSalamanca in September 2003, has been written by Fernando Sancho. AppendixD, an introduction to stability conditions for derived categories, has been writtenby Emanuele Macrı. We are deeply thankful to Emanuele and Fernando for theircontributions to this work.

Very special thanks are due to David Ploog, who was patient enough to readthis book twice during the final stage of its redaction, pointed out several mistakesand inconsistencies, and proposed several improvements. Without David’s help thisbook would definitely have been worse. We also thank Benoit Charbonneau, Ste-fano Guerra, Daniel Hernandez Serrano, Adrian Langer, Cristina Lopez Martın,Tony Pantev, Darıo Sanchez Gomez, Justin Sawon, Edoardo Sernesi, Carlos TejeroPrieto and the anonymous referee for pointing out mistakes and suggesting im-provements.

We have included many results obtained jointly with our collaborators BjornAndreas, Antony Maciocia, Jose M. Munoz Porras and Fabio Pioli: we are gratefulto all of them. We also thank Bjorn for clarifying some physics-related issues.

Many thanks are due to Birkhauser, especially in the person of Ann Kostant,for the enthusiasm with which this project was considered, and for patiently wait-ing until it was over.

Writing this book has required many exchanges of visits among the authors(very likely, the most pleasant aspect of the enterprise). These have been madepossible by funding provided by the University of Genova, the International Schoolfor Advanced Studies in Trieste and the University of Salamanca, by the Italian

xvi Acknowledgments

National Research Projects (P.R.I.N.) “Geometric methods in the theory of nonlin-ear waves” and “Geometry on algebraic varieties,” and by the Spanish ResearchProjects “Geometrical integral transforms and applications” (BFM2003-00097),“Applications of integral functors to geometry and physics” (GCyL-SA114/04)and “Coherent sheaves, derived categories and integral functors in birational ge-ometry and string theory” (MTM2006-04779). A strong initial impulse to thewriting of the book was given by a “Research in Pairs” stay of the first and secondauthor at Mathematische Forschunginstitut Oberwolfach, which we deeply thank,especially in the person of its director, Professor Martin Greuel. The editing of thebook was finalized when the second author was spending the 2008 spring term atthe Department of Mathematics of the University of Pennsylvania in Philadelphia.Warm thanks are due to the scientists and staff of the department for their collab-oration and the nice atmosphere they create and to the university for providingsupport.

Chapter 1

Integral functors

Introduction

The first instance of an integral functor is to be found in Mukai’s 1981 paper onthe duality between the derived categories of an Abelian variety and of its dualvariety [224]. Integral functors have also been called “Fourier-Mukai functors” or“Fourier-Mukai transforms.” However, we shall give these terms specific meaningsthat we shall introduce in Chapter 2.

The core idea in the definition of an integral functor is very simple: if wehave two varieties X and Y , we may take some “object” on X, pull it back to theproduct X×Y , twist it by some object (“kernel”) in X×Y and then push it downto Y (i.e., we integrate on X). This is what happens with the Fourier transformof functions: one takes a function f(x) on Rn, pulls it back to Rn×Rn, multipliesit by the kernel ei x·y and then integrates over the first copy of Rn, thus obtaininga function f(y) on the second copy.

is the reason for the “Fourier” appearing in the original name of these functors (thechristening was done by Mukai in [224]). But the analogy goes further, since forintegral functors one can talk (under suitable conditions) about an inverse functor,convolutions (composing two such functors amounts to convolute the kernels in asuitable way), a Parseval theorem (which states that, in the appropriate sense, an

Technically, if we denote by D−(X) the derived categories of complexes ofcoherent sheaves on X that are bounded on the right (here X and Y are properalgebraic varieties over a field k), an integral functor is a functor

ΦK•

X→Y : D−(X)→ D−(Y )

1

This naive resemblance between an integral functor and the Fourier transform

Progress in Mathematics 276, DOI: 10.1007/b11801_1,C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics,

integral functor is an “isometry”), etc.

© Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

2 Chapter 1. Integral functors

of the formΦK•

X→Y (E•) = RπY ∗(π∗XE•L⊗K•)

where K• is an object in D−(X × Y ). The morphisms πX , πY are projectionsonto the factors of the Cartesian product. Our aim in this chapter is to developa fairly general theory of such functors, without making detailed assumptions onthe varieties X and Y and on the kernel K•. In Chapter 2 we shall consider thecase of integral functors that are exact equivalences of categories. More specificsituations (e.g.,X and Y Abelian varieties, or K3 surfaces, or hyperkahler varieties,in each case with a kernel of appropriate type, etc.) will be studied in the followingchapters.

Let us briefly describe the structure of this chapter. After fixing in Section1.1 some notation, conventions and definitions, in the next section we introducethe integral functors and study their first properties (base change and adjointfunctors). In Section 1.3 we prove a criterion for the testing whether an integralfunctor is fully faithful. Finally we introduce equivariant integral functors. Thiswill be used in Chapter 7 to study the McKay correspondence.

1.1 Notation and preliminary results

By algebraic variety X we mean a separated scheme of finite type over an alge-braically closed field k. We do not assume a priori that X is irreducible or reduced.The symbol D(X) will denote the derived category of complexes of quasi-coherentOX -modules with coherent cohomology sheaves. (Details about the construction ofthe category D(X) are given in Appendix A; here we only set the basic notation.)We denote by Db(X), D+(X) and D−(X) the corresponding derived categories ofbounded complexes, bounded on the left and bounded on the right complexes, re-spectively; their objects will be denoted by such symbols as E•, and the set of mor-phisms between two objects in one of these derived categories by HomD(X)(E•,F•).It is a remarkable fact that Db(X) is equivalent to the bounded derived categoryof the Abelian category of coherent sheaves on X. If f : X → Y is a proper mor-phism to another projective variety, we shall denote by Rf∗ : D+(X) → D+(Y ),Lf∗ : D−(Y ) → D−(X) the associated derived functors (properness is necessarybecause otherwise the direct image of a coherent sheaf may fail to be coherent).Since every sheaf has only a finite number of nonvanishing higher direct images andthis number is bounded by the maximum of the dimensions of the fibers, the de-rived direct image can be extended to a functor Rf∗ : D(X)→ D(Y ) and inducesfunctors Rf∗ : Db(X) → Db(Y ), Rf∗ : D−(X) → D−(Y ). If the morphism f hasfinite Tor-dimension, that is, if there is a number that bounds the number of non-vanishing higher inverse images for every sheaf on Y , then Lf∗ : D−(Y )→ D−(X)extends to a functor Lf∗ : D(Y )→ D(X) and induces Lf∗ : Db(Y )→ Db(X).

1.1. Notation and preliminary results 3

If the morphism f is flat, the inverse image functor is exact, hence it doesnot need to be derived, so that one actually has Lf∗ = f∗. Analogously, if f is anaffine morphism (for instance, a closed immersion), the direct image functor doesnot need to be derived, and we have Rf∗ = f∗.

The bifunctor induced by the left derived functor of the tensor product will be

denoted byL⊗ . One can also derive the homomorphism sheaf bifunctor, obtaining

the functor RHom•OX (E•,F•) (for either E• bounded on the right or F• boundedon the left, see Appendix A for details). For any object M• in D(X) we have a“dual object” in the derived category D(X), M•∨ = RHom•OX (M•,OX). Notethat when M• is a complex concentrated in degree zero, the cohomology sheavesof M•∨ are the local Ext sheaves, Hi(M•∨) = ExtiOX (M0,OX×Y ).

So we shall use the symbol ∨ for the dual in derived category, while for thedual of a sheaf we shall use the notation ∗ (but for the dual of a line bundle L weshall sometimes also write L−1).

Assume now that the variety X is smooth. Then any complexM• in Db(X)is isomorphic in the derived category to a bounded complex E• of locally freesheaves, i.e., to a perfect complex (cf. Definition A.42). In this situation the dualof an object of Db(X) is still an object of Db(X), and, remarkably, all objects in thebounded derived category are “reflexive,” that is,M• ' (M•∨)∨ (see PropositionsA.75 and A.87).

If X is also proper, the Chern characters of an object M• of Db(X) aredefined by

chj(M•) =∑i

(−1)i chj(E i) ∈ Aj(X)⊗Q ,

where Aj(X) is the degree-j summand of the Chow ring (when k = C, the groupAj(X) ⊗ Q is the algebraic part of the rational cohomology group H2j(X,Q)).This definition is well posed since it is independent of the choice of the boundedcomplex E• of locally free sheaves. By definition the rank of M• is the integernumber rk(M•) = ch0(M•).

For a complex E• in Db(X) we define its Mukai vector as

v(E•) = ch(E•) ·√

td(X) , (1.1)

where td(X) ∈ A•(X)⊗Q is the Todd class of X.

There is a natural involution on the Chow ring ∗ : A•(X)→ A•(X) which actson the degree-j summand Aj(X) as the multiplication by (−1)j . Given v ∈ A•(X),the element v∗ is called the Mukai dual of v. If E• is an object of Db(X), one hasch(E•)∗ = ch(E•∨). Due to the identity√

td(X) = (√

td(X))∗ · exp( 12c1(X))


it turns out thatv(E•∨) = v(E•)∗ · exp( 1

2c1(X)) . (1.2)

In particular, for a locally free sheaf E one has v(E∗) = v(E)∗ · exp( 12c1(X)).

We define on the rational Chow group A•(X)⊗Q a symmetric bilinear form〈·, ·〉, called Mukai pairing, by setting

〈v, w〉 = −∫X

v∗ · w · exp( 12c1(X)) . (1.3)

We shall write v2 instead of 〈v, v〉. When k = C, the involution ∗ and the Mukaipairing naturally extend to the even rational cohomology ⊕jH2j(X,Q).

When X is a surface we can write a Mukai vector v in the form v = (v0, v1, v2)with vj ∈ Aj(X)⊗Q, and the Mukai pairing takes the form

〈v, w〉 = v1 · w1 − v0 · (w2 − 12w1 · c1(X))−w0 · (v2 − 1

2v1 · c1(X))− 14v0 · w0 · c1(X)2 . (1.4)

When the first Chern class of the surface is trivial (X is K3 or Abelian), the Mukaipairing takes the form

〈v, w〉 = −∫X

v∗ · w = v1 · w1 − v0 · w2 − w0 · v2 .

We shall make use of these expressions mainly in Chapter 4.

As a first application of the notion of Mukai vector, we use it to express theEuler characteristic of two objects E• and F• of the bounded derived categoryDb(X) of a smooth proper variety X. This is defined as

χ(E•,F•) =∑i

(−1)i dim HomiD(X)(E•,F•) (1.5)

where HomiD(X)(E•,F•) = HomD(X)(E•,F•[i]), cf. eq. (A.12). When E and F are

sheaves, one has HomiD(X)(E ,F) ' ExtiX(E ,F), see Proposition A.68. The Euler

characteristic χ(F) of a sheaf coincides with χ(OX ,F).

Note that the Grothendieck-Hirzebruch-Riemann-Roch theorem gives the for-mula

χ(E•,F•) =∫X

ch(E•∨) · ch(F•) · td(X) = −〈v(E), v(F)〉 . (1.6)

When X is a surface, the Euler characteristic of E• and F• can be computedusing eq. (1.6) by means of the explicit expression (1.4) of the Mukai pairing. Inparticular, if rk(E•) = rk(F•) = 0 (when E• and F• are sheaves this means thatthey are torsion sheaves), we obtain

χ(E•,F•) = −c1(E•) · c1(F•) . (1.7)

1.2. First properties of integral functors 5

The symbol Ox will denote as usual the skyscraper sheaf of length 1 at theclosed point x ∈ X: this is the structure sheaf of x as a closed subscheme of X.These sheaves satisfy the following properties:

HomiD(X)(Ox1 ,Ox2) = 0 for all i ∈ Z if x1 6= x2 ; (1.8)

HomiD(X)(Ox,Ox) '

∧iTxX for 0 ≤ i ≤ dimX

0 otherwise(1.9)

where TxX is the tangent space to X at x. This is proved by resolving Ox locallyby the Koszul complex.

1.2 First properties of integral functors

Let X, Y be proper algebraic varieties over k; the projections of the Cartesianproduct X × Y onto the factors X, Y are denoted by πX , πY respectively. Let K•be an object in the derived category D−(X × Y ). We define the functor

ΦK•

X→Y : D−(X)→ D−(Y )

by letting

ΦK•

X→Y (E•) = RπY ∗(π∗XE•L⊗K•) .

The complex K• will be called the kernel of the functor, and ΦK•

X→Y will be calledthe associated integral functor.

Remark 1.1. One may also define integral functors D(Qco(X)) → D(Qco(Y ))(with a kernel inD(Qco(X×Y ))) because the derived tensor product is well definedalso for unbounded complexes of quasi-coherent sheaves (see Section A.4.5). 4

Example 1.2. Let δ : X → X×X denote the diagonal immersion, and write ∆ forits image. Let K• be the complex in Db(X ×X) concentrated in degree zero withK0 = O∆ = δ∗OX . It is easy to check that ΦK

•

X→X is isomorphic to the identityfunctor. More generally, if L is a line bundle on X, the functor Φδ∗LX→X acts as thetwist by L. Let us also note that, given a proper morphism f : X → Y , by takingas K• the structure sheaf of the graph Γf ⊂ X × Y , one has isomorphisms offunctors ΦK

•

X→Y ' Rf∗ and ΦK•

Y→X ' Lf∗. 4

A useful feature of integral functors is that the composition of two of them isstill an integral functor, whose kernel may be expressed as a kind of “convolutionproduct” of the two original kernels. This may be seen as an analogue of thecomposition law for correspondences [209]. Let us describe this property. If Z is


another proper variety, we consider the diagram

X × Y × ZπXY

xxppppppppppπY,Z

πXZ

&&NNNNNNNNNN

X × Y Y × Z X × Z

Given kernels K• in D−(X × Y ) and L• in D−(Y ×Z), we define the convolutionof L• and K• as the kernel in D−(X × Z)

L• ∗ K• = RπXZ∗(π∗XYK•L⊗π∗Y ZL•) .

Proposition 1.3. There is a natural isomorphism of functors

ΦL•

Y→Z ΦK•

X→Y ' ΦK•∗L•

X→Z .

Proof. Given E• in D−(X), one has:

ΦL•

Y→Z(ΦK•

X→Y (E•)) = RπZ∗(π∗Y [RπY ∗(π∗XE•L⊗K•)]

L⊗L•)

' RπZ∗(RπY,Z∗(π∗XY (π∗XE•L⊗K•))

L⊗L•)

' RπZ∗(RπY,Z∗(π∗XY (π∗XE•L⊗K•)

L⊗π∗Y ZL•))

' RπZ∗RπXZ,∗(π∗XZπ∗XE•

L⊗π∗XYK•

L⊗π∗Y ZL•)

' RπZ,∗(π∗XE•L⊗ (L• ∗ K•)) .

The isomorphism in the second line is cohomology flat base change (PropositionA.85), those in the third and fifth lines are the projection formula in derivedcategory (Proposition A.83), while in the fourth line the isomorphism is due tothe obvious identities πX πXY = πX πXZ and πZ πXZ = πZ πY Z .

A natural issue to be addressed is when a kernel defines an integral functorwhich maps Db(X) to Db(Y ). We say that a kernel K• in D−(X × Y ) is of finiteTor-dimension as a complex of OX-modules if it is isomorphic in the derivedcategory to a bounded complex of sheaves that are flat over X.

Proposition 1.4. Let K• an object of D−(X × Y ) of finite Tor-dimension as acomplex of OX-modules. The integral functor ΦK

•

X→Y has the following property:there exist integer numbers z and n ≥ 0 such that for every coherent sheaf F onX, the cohomology sheaves Hi(ΦK•X→Y (F)) vanish for i /∈ [z, z + n].

Proof. Take a bounded complex E• ≡ Ez → Ez+1 → · · · → Ez+m of flat sheavesover X isomorphic to K• in the derived category. If r is the dimension of X,then for every coherent sheaf F on X the cohomology sheaves Hi(ΦK•X→Y (F)) =Hi(RπY ∗(π∗XF ⊗ E•)) vanish unless z ≤ i ≤ z + n with n = m+ r.


Corollary 1.5. If K• is of finite Tor-dimension as a complex of OX-modules, thefunctor ΦK

•

X→Y maps Db(X) to Db(Y ).

Note that the condition of finite Tor-dimensionality is always fulfilled if Xand Y are smooth and K• is an object of Db(X × Y ), because in this case K• isisomorphic in the derived category to a bounded complex of locally free sheavesand the projection πX is a flat morphism.

Another important feature of integral functors is that they are exact as func-tors of triangulated categories (see Definition A.46 and Proposition A.62), sincethey are compositions of exact functors: derived tensor product, inverse image un-der a flat morphism, and direct image in derived category. In particular, for anyexact sequence 0→ F → E → G → 0 of coherent sheaves in X we obtain an exactsequence

· · · → Φi−1(G)→ Φi(F)→ Φi(E)→ Φi(G)→ Φi+1(F)→ . . . (1.10)

where we have written Φi(•) = Hi(ΦK•X→Y (•)) for simplicity.

In the next chapters we shall be often interested in studying cases where anintegral functor applied to a complex or a sheaf yields a concentrated complex.

Definition 1.6. Given an integral functor ΦK•

X→Y , a complex F• in D−(X) satisfiesthe WITi condition (or is WITi) if there is a coherent sheaf G on Y such thatΦK•

X→Y (F•) ' G[−i] in D(Y ), where G[−i] is the associated complex concentratedin degree i. We say that F• satisfies the ITi condition if in addition G is locallyfree. 4

The acronym “IT” stands for “index theorem” and “W” stands for “weak.”The reason for this terminology (which is not entirely appropriate in the mostgeneral case of nonconcentrated complexes) will be made clear in Chapter 5, wherea link between the integral functors and index theory is established.

Proposition 1.7. Assume that the kernel is a locally free sheaf Q on the product.A coherent sheaf F on X is ITi if and only if Hj(X,F ⊗ Qy) = 0 for all y ∈ Yand for all j 6= i, where Qy denotes the restriction of Q to X ×y. Furthermore,F is WIT0 if and only if it is IT0.

Proof. Both statements follow from the cohomology base change theorem [141,III.12.11] taking into account that π∗XF ⊗Q is flat over Y .

If X and Y are smooth proper varieties, and K• is a kernel in Db(X×Y ), bythe Grothendieck-Riemann-Roch theorem the integral functor ΦK

•

X→Y : Db(X) →


Db(Y ) gives rise to the commutative diagram

Db(X)ΦK•

X→Y //

v

Db(Y )

v

A•(X)⊗Q

fK•

// A•(Y )⊗Q

(1.11)

where fK•

is the homomorphism of Q-vector spaces defined by

fK•(α) = πY ∗(π∗Xα · v(K•)) (1.12)

and v(K•) is the Mukai vector (1.1) of K•. Notice that

v(K•) = π∗X√

tdX · ch(K•) · π∗Y√

tdY .

The map fK•

depends functorially on the kernel, i.e.,

fK•∗L• = fL

• fK

•. (1.13)

When the base field k is the field C of the complex numbers, we can extendthe diagram (1.11) to a diagram

Db(X)ΦK•

X→Y //

v

Db(Y )

v

H•(X,Q)

fK•

// H•(Y,Q)

(1.14)

where the homomorphism in the bottom line is defined by a formula like (1.12)and maps the even cohomology ring H2•(X,Q) to H2•(Y,Q). The analogue ofEquation (1.13) holds true.

1.2.1 Base change formulas

We wish to generalize the notion of integral functor ΦK•

X→Y : D−(X) → D−(Y )to “relative” integral functors D−(T × X) → D−(T × Y ) where T is a (proper)variety of “parameters.” Although in further chapters we shall extend this notionto nontrivial families, we stick for the time being to this simple situation becauseit is general enough for our present purposes.

For any variety T we shall write XT = T×X and denote by πT the projectionXT = T ×X → T , unless confusion can arise. Given a kernel K• ∈ D−(X × Y )


we define K•T = π∗X×YK• and consider the functor ΦT = ΦK•T

XT→YT : D−(XT ) →D−(YT ) given by

ΦT (E•) = RπYT ∗(π∗XT E

•L⊗K•T ) ,

where

πXT : (X × Y )T = T ×X × Y → XT = T ×XπYT : (X × Y )T = T ×X × Y → YT = T × Y

πX×Y : (X × Y )T = T ×X × Y → X × Y

are projections. The functor ΦT can be regarded as an integral functor with kerneli∗K•T in D−(XT ×YT ), where i : XT ×T YT → XT ×YT is the closed immersion ofthe fiber product. The notions of WITi and ITi introduced in Definition 1.6 applyto this new situation.

What makes relative integral functors interesting is their compatibility withbase changes. Let f : S → T be a morphism and denote by fZ the induced mor-phism S × Z → T × Z for any Z.

Proposition 1.8. For every object E• in D−(T ×X) there is a functorial isomor-phism

Lf∗Y ΦT (E•) ' ΦS(Lf∗XE•)in the derived category of YS.

Proof. One has

Lf∗Y ΦT (E•) = Lf∗YRπYT ∗(π∗XT E

•L⊗K•T ) ' RπYS∗Lf

∗X×Y (π∗XT E

•L⊗K•T )

by base change in the derived category (Proposition A.85). Then

Lf∗X×Y (π∗XT E•

L⊗K•T ) ' π∗XS (Lf∗XE•)

L⊗K•S ,

whence the statement follows.

Note that we do not need to assume that the base change morphism f : S → T

is flat.

If the original kernel K• is of finite Tor-dimension as a complex of OX -modules, then K•T is of finite Tor-dimension as a complex of OXT -modules so thatProposition 1.4 implies that ΦT is bounded and can be extended to a functor ΦT :D(T ×X)→ D(T × Y ) for every T and maps Db(T ×X)→ Db(T × Y ).

Base change compatibility means that if we think of an object E• ∈ D(T×X)as a family of objects Lj∗t E• ∈ D(X), then the relative integral functor ΦT (E•) isthe family of integral functors Φt(Lj∗t E•), that is,

Lj∗t ΦT (E•) ' Φt(Lj∗t E•) . (1.15)


We shall often be interested in transforming a complex that reduces to asingle sheaf E on T×X. We can derive from the base change (1.15) the relationshipbetween the sheaves ΦiT (E)t = j∗t ΦiT (E) and the sheaves Φit(Et), where Et = j∗t E .Note that ΦiT (E) = 0 for every sheaf E when i > n = dimX +m, where Hm(K•)is the highest nonzero cohomology sheaf of the kernel.

Corollary 1.9. Let E be a sheaf over XT = T × X. The formation of ΦnT (E) iscompatible with base change for sheaves, that is, one has

ΦnT (E)t ' Φnt (Et)

for every point t ∈ T . Moreover, if E is flat over T , the following results hold true.

1. For every point t in T there is a convergent spectral sequence

E−p,q2 (t) = TorOTp (ΦqT (E),Ot) =⇒ Eq−p∞ (t) = Φq−pt (Et) .

2. Assume that E is WITi and write E = ΦiT (E). Then for every t ∈ T thereare isomorphisms of sheaves over Xt

TorOTj (E ,Ot) ' Φi−jt (Et) , j ≤ i .

3. The restriction Et to the fiber Xt is WITi for every t ∈ T if and only if Eis WITi and E = ΦiT (E) is flat over T . In that case the formation of E iscompatible with base change, that is, (E)t ' Et for every point t ∈ T .

Proof. Let us write Lpf∗t (F) = H−p(Lf∗t (F)) for any sheaf F . For every pointt ∈ B there exist two spectral sequences E−p,q2 (t) = Lpj

∗t (ΦqT (E)), E−p,q2 (t) =

Φq(Lpj∗E) converging respectively to Eq−p∞ (t) = Hq−p(Lj∗t ΦT (E) and Eq−p∞ (t) =Hq−p(Φt(Lj∗t E)). One deduces isomorphisms ΦnT (E)t ' En∞(t) and Φnt (Et) 'En∞(t), so that ΦnT (E)t ' Φnt (Et) by Proposition 1.8. For the rest of the proof,E is flat over T . Then the spectral sequence E−p,q2 (t) degenerates and one hasstatements 1 and 2. Regarding 3, the only point worthy of a comment is that if Etis WITi for every t, then E is WITi and E = Φi(E) is flat over T . Let q0 be themaximum of the q’s with Φq(E) 6= 0. Then E0,q0

2 (t) = Φq0(E)⊗OT Ot 6= 0 for somepoint t. Since E−2,q0+1

2 (t) = 0, every nonzero cycle in the term E0,q02 (t) survives

at infinity, whence Eq0∞(t) 6= 0 and then q0 = i. The same argument proves thatEp,i2 (t) = 0 for every point t and every p > 0, so that E is flat over T . SinceE−2,i

2 (t) = 0, any nonzero cycle E0,i−12 (t) survives at infinity as before; this im-

plies Ei−1∞ (t) 6= 0 which is absurd. Then Φi−1(E)⊗OT Ot = E0,i−1

2 (t) = 0 for everypoint t so that Φi−1(E) = 0. Proceeding as above one proves that Φj(E) = 0 forevery j < i. The last part follows now from Equation (1.15).


The following formula will be used to prove that in many cases the kernel K•is uniquely determined, up to isomorphism in the derived category, by the integralfunctor ΦK

•

X→Y .

Proposition 1.10. Let Φ = ΦK•

X→Y an integral functor with kernel K•. Then

K• ' ΦX(O∆)

where ΦX is the integral functor from D−(X×X) to D−(X×Y ) with kernel K•X .

Proof. Let δ : X → X ×X the diagonal immersion and δY : X ×Y → X ×X ×Ythe induced immersion. Then,

ΦX(O∆) ' RπYX∗(π∗XX δ∗(OX)

L⊗K•X) ' RπYX∗(δY ∗π

∗XOX×X

L⊗K•X)

' RπYX∗(δY ∗(δ∗YK•X)) ' K•

where the second isomorphism is base change and the third is the projectionformula.

The following result can be found in [61, Lemma 4.3]. Let us denote by jtthe immersion X → T ×X as the fiber Xt = π−1

T (t)).

Proposition 1.11. Let F• be an object in Db(YT ). If the restriction Lj∗tF• is aconcentrated complex Ft for every (closed) point t ∈ T , then F• is concentratedas well, and is a coherent sheaf F on YT , flat on T , and such that j∗tF ' Ft forevery t ∈ T .

Proof. For every t ∈ T there is a spectral sequence E−p,q2 = Lpj∗tHq(F•) converg-

ing to Eq−p∞ = Hq−p(Lj∗tF•). Let q0 be the maximum of the indexes q such thatHq(F•) 6= 0. Then for some t ∈ T one has E0,q0

2 6= 0 and E0,q02 survives to infinity

so that Hq0(Lj∗tF•) 6= 0. Then q0 = 0. Now any nonzero element in E−1,02 survives

to infinity and since E−1∞ = 0, one has E−1,0

2 = 0 which implies that F = H0(F•)is flat over T . Thus E−p,02 = 0 for every p ≥ 1. Finally, if q1 < 0 is the largeststrictly negative index q with Hq(F•) 6= 0, then E0,q1

2 6= 0 for some t and E0,q12

survives to infinity so that Hq1(Lj∗tF•) 6= 0; this is absurd, hence no such q1 existsand F• ' F in the derived category.

We can easily derive a simple consequence of this fact.

Corollary 1.12. Let K• ∈ Db(X × Y ) be of finite Tor-dimension over X and letΦK•

X→Y : Db(X) → Db(Y ) be the corresponding integral functor. Assume that for


every (closed) point x ∈ X one has ΦK•

X→Y (Ox) ' Oy for some (closed) pointy ∈ Y . Then there is a morphism f : X → Y and a line bundle L on X such that

ΦK•

X→Y (E•) ' Rf∗(E• ⊗ L)

for every object E• in Db(X).

Proof. Since Lj∗xK• ' ΦK•

X→Y (Ox) ' Oy, by Proposition 1.11 the kernel K• reducesto a single sheaf K on X×Y flat over X. Moreover K is a flat family of skyscrapersheaves on Y of length 1. Since Y is a fine moduli space for its own points andthe diagonal ∆ → Y × Y is a universal family, there is a morphism f : X → Y

such that K ' (f × 1)∗(O∆)⊗ π∗XL for some line bundle L on X. Now the proofis completed by a simple computation.

1.2.2 Adjoints

The issue of the existence of right and left adjoint functors of an integral functorcan be naturally addressed within the framework of the Grothendieck-Serre theoryof duality [139, 291], which we describe in Section C.1.

We shall use indeed Grothendieck-Serre duality to prove that under quitegeneral conditions, integral functors have left and right adjoints.

Proposition 1.13. Let X, Y be proper algebraic varieties, of dimensions m and n,and let K• be a kernel in Db(X × Y ).

1. If K• is of finite Tor-dimension over Y and X is smooth, the functor

ΦK•∨⊗π∗XωX [m]

Y→X : Db(Y )→ Db(X)

is right adoint to the functor ΦK•

X→Y .

2. If K• is of finite Tor-dimension over X and Y is smooth, the functor

ΦK•∨⊗π∗Y ωY [n]

Y→X : Db(Y )→ Db(X)

is left adjoint to the functor ΦK•

X→Y .

Proof. In both cases, K• is of finite Tor-dimension over X×Y , and the same is true

for the derived dual K•∨. As a consequence, (−)L⊗K•∨ is both left and right adjoint

to (−)L⊗K• as functors on Db(X × Y ) by Corollary A.88. We only prove the first

part, since the second is analogous. Since ΦK•

X→Y = RπY ∗ ((−)L⊗K•) π∗X , a right

adjoint to ΦK•

X→Y is the composition in the reverse order of the right adjoints to the

factors, namely, RπX∗ ((−)L⊗K•) [((−)⊗π∗XωX [m])π∗Y ] = ΦK

•∨⊗π∗XωX [m]Y→X .


Remark 1.14. If X and Y are smooth and proper with the same dimension andthe canonical bundles ωX and ωY are trivial, the left and right adjoint coincide.

4

Let X be a smooth proper variety. Since any object on Db(X) has finiteTor-dimension, applying Equation (C.13) we have that the functor

SX : Db(X)→ Db(X)

F• 7→ F• ⊗ ωX [n]

provides a natural isomorphism

HomD(X)(F•,G•) ' HomD(X)(G•, SX(F•))∗ (1.16)

where F• and G• are complexes in Db(X).

Proposition 1.15. If X and Y are smooth and proper, for every complex F• inDb(X) there is an isomorphism (ΦK

•

X→Y (F•))∨ ' ΦK•∨

Y→X(SX(F•∨)).

Proof. This follows from the chain of isomorphisms

(ΦK•

X→Y (F•))∨ = RHom•OY (RπY ∗(π∗XF•L⊗K•),OY )

' RπY ∗RHom•OX×Y (π∗XF•L⊗K•, π∗XωX [m])

' RπY ∗((π∗XF•L⊗K•)∨ ⊗ π∗XωX [m])

' RπY ∗(π∗X(F•∨ ⊗ ωX [m])L⊗K•∨)

' ΦK•∨

Y→X(F•∨ ⊗ ωX [m]) = ΦK•∨

Y→X(SX(F•∨))

where the first isomorphism is duality for πY .

The isomorphism (1.16) fits into a commutative diagram

HomD(X)(F•,G•)

SX **UUUUUUUUUUUUUUUU// HomD(X)(G•, SX(F•))∗

HomD(X)(SX(F•), SX(G•)) .

(1.17)

An easy consequence of Serre duality is the following formula involving theEuler characteristic of two bounded complexes (see Eq. (1.5))

χ(E•,F•) = χ(F•, E• ⊗ ωX) , (1.18)

where E• and F• are in Db(X).


The functor SX is the model for an abstract definition of Serre functor fortriangulated categories [45].

We remind that a functor F : A→ B is an equivalence of categories when ithas a quasi-inverse functor, that is a functor G : B → A such that G F ' IdA

and F G ' IdB. In this case G is both a left and a right adjoint to F .

Definition 1.16. Let A be a k-linear triangulated category whose sets of homomor-phisms are finite-dimensional vector spaces. An exact autoequivalence SA : A→ A

is a Serre functor if for all a, b in A there is a bifunctorial isomorphism

HomA(a, b) ' HomA(b, SA(a))∗ .

4

A Serre functor automatically satisfies a compatibility condition expressedby a diagram analogous to (1.17). A Serre functor, if it exists, is unique up tofunctorial isomorphisms.

Serre functors provide a handy tool for describing adjoints of exact functors.

Lemma 1.17. Let F : A → B be an exact functor between triangulated categorieswith Serre functors SA and SB. Then F has a left adjoint G if and only if it hasa right adjoint H and one has H = SA G S−1

B .

Proof. We prove the statement in one direction, the other being analogous. If Gis a left adjoint to F , then

HomA(a, SA G S−1B (b)) ' HomA(G S−1

B (b), a)∗ ' HomB(S−1B (b), F (a))∗

' HomB(b, SB F (a))∗ ' HomB(F (a), b)

so that H = SA G S−1B is a right adjoint to F .

In particular, when F is an equivalence of triangulated categories, since itsleft and right adjoints coincide, we get the following result.

Corollary 1.18. Let F : A → B be an exact equivalence of triangulated categorieswith Serre functors SA, SB. Then SB F ' F SA.

Remark 1.19. If X is a proper Gorenstein variety, one can define the functorSX : Db(X) → Db(X) by letting SX(F•) = F• ⊗ ωX [n], where n = dimX asin the smooth case. However, this is a Serre functor for Db(X) in the sense ofDefinition 1.16 only if X is smooth. Indeed, if SX is a Serre functor, for every pointx ∈ X, every i ∈ Z and every coherent sheaf M on X, one has isomorphisms

ExtiX(Ox,M) ' Extn−iX (M,Ox ⊗ ωX)∗ ' Extn−iX (M,Ox)∗ .

Then ExtiX(Ox,M) = 0 for all i > n and any M, so that Ox has finite Tor-dimension. Hence x cannot be a singular point. 4

1.3. Fully faithful integral functors 15

We have proved explicitly that integral functors admit right and left adjoints.One should note however that every exact functor Db(X) → Db(Y ) between thebounded derived categories of smooth projective varieties has a right adjoint (andhence, since both categories have Serre functors, a left adjoint as well). This fol-lows from the fact that these categories are saturated. Let us recall this result.Let A be a triangulated category. A cohomological functor K from A to the cat-egory of k-vector spaces Vect(k) is said to be of finite type if for all a ∈ A, thesum

∑i dimk K(a[i]) is finite (the notion of cohomological functor is defined in

Appendix A). Moreover A is said to be saturated if every cohomological functor offinite type is representable (the notions of cohomological and representable func-tors are given in Appendix A, see Definitions A.47 and A.1). A remarkable resultby Bondal and Van den Bergh [52] states that for any smooth projective varietyX, the bounded derived category Db(X) is saturated.

Proposition 1.20. Let X and Y be smooth projective varieties. Every exact functorF : Db(X)→ Db(Y ) has a right and a left adjoint.

Proof. For every complex G• in Db(Y ), the functor K : Db(X) → Vect(k) givenby K(E•) = HomDb(Y )(F (E•),G•) is a cohomological functor of finite type, sothat it is representable by an object M• in Db(X), unique up to isomorphisms.By Yoneda’s lemma, by setting H(G•) =M• one defines a functor H : Db(Y ) →Db(X) which is right adjoint to F . Lemma 1.17 implies the existence of a leftadjoint as well.

1.3 Fully faithful integral functors

Our next step is to establish a criterion for an integral functor to be fully faithful.This will be expressed by Theorem 1.27.

1.3.1 Preliminary results

We recall that a functor F : A→ B is fully faithful if the map

F : HomA(a1, a2)→ HomB(F (a1), F (a2))

is bijective for any pair of objects a1, a2 in A.

If the functor F : A → B has a left adoint G, then F is fully faithful if andonly if the natural adjunction map γ : GF → IdA is an isomorphism. Analogously,if F has a right adjoint H, it is fully faithful if and only if the natural adjunctionmap η : IdA → H F is an isomorphism. Then, given an object a in A, theinduced morphism γ(a) : G(F (a)) → a is zero if and only if F (a) = 0. Similarly,η(a) : a→ H(F (a)) is zero if and only if F (a) = 0.


Remark 1.21. One can state a handy criterion for checking if a functor F : A→ B

is fully faithful. If F has a left adoint G, there is a commutative diagram

HomA(a1, a2) F //

GF

HomB(F (a1), F (a2))

'

HomA((G F )(a1), (G F )(a2))γ(a2) // HomA((G F )(a1), a2) .

We can derive two consequences of this. The first is that if the map

G F : HomA(a1, a2)→ HomA((G F )(a1), (G F )(a2)

is injective, then the map

HomA(a1, a2)→ HomB(F (a1), F (a2))

is injective as well. The second is that if G F is fully faithful, so that both thefirst vertical arrow and the bottom arrow are isomorphisms, then F is also fullyfaithful. 4

To prove Theorem 1.27 we need a preliminary lemma which characterizesobjects of the derived category supported on a closed subvariety.

Lemma 1.22. [48, Prop. 1.5] Let j : Y → X be a codimension d closed immersionof irreducible smooth algebraic varieties and K• an object of Db(X). Assume that

1. Lj∗xK• = 0 for every (closed) point x ∈ X − Y , where jx : x → X denotesthe embedding of x into X;

2. Lij∗xK• = 0 when either i < 0 or i > d for every (closed) point x ∈ Y .

Then there is a sheaf on X whose topological support is Y which is isomorphic toK• in Db(X).

Proof. Recall that Lij∗xK• denotes the cohomology sheaf H−i(j∗xL•) where L• isa bounded complex of flat sheaves quasi-isomorphic to K•. Let us write Hq =Hq(K•). For every point x ∈ X there is a spectral sequence

E−p,q2 = Lpj∗xHq =⇒ E−p+q∞ = Lp−qj

∗xK• .

If q0 is the maximum of the q’s with Hq 6= 0, then a nonzero element in j∗xHq0survives up to infinity in the spectral sequence. Since Eq∞ = L−qj

∗xK• = 0 for every

x ∈ X and q > 0, one has q0 ≤ 0. We now show that the topological support of allthe sheaves Hq is contained in Y . Assume indeed that this is not true and considerthe maximum q1 of the q’s such that j∗xHq 6= 0 for a certain point x ∈ X − Y ;


then a nonzero element in j∗xHq1 survives up to infinity in the spectral sequence,which is absurd since Lj∗xK• = 0.

Let q2 ≤ q0 be the minimum of the q’s with Hq 6= 0. We know that Hq2is topologically supported on a closed subset of Y . Take a component Z ⊆ Y ofthe support. Let d be the codimension of Y in X. If c ≥ d is the codimension ofZ, then, for every x ∈ Z outside a closed subset of Z of codimension greater orequal to c+ 2 in X, one has Lpj∗xHq2+1 = 0 for every p ≥ c+ 2 and Lcj∗xHq2 6= 0.This follows from the fact that the m-singularity set of a coherent sheaf is a closedsubscheme of dimension smaller than or equal to m; this is proved in [184, Thm.5.8] in the complex case, while for a proof in the general case, the reader mayrefer, e.g., to [144, Prop. 1.13]. For such a point x ∈ Z any nonzero element ofLcj∗xHq2 survives in the spectral sequence up to infinity and gives Lc−q2j

∗xK• 6= 0.

Thus c− q2 ≥ 0 which implies q2 ≥ c− d ≥ 0 and then q2 = q0 = 0. So K• ' H0

in Db(X). We already know that the topological support of H0 is contained inY ; actually it is the whole of Y : if this were not true, since Y is irreducible, thesupport would have a component Z ⊂ Y of codimension c > d and one could find,by reasoning as above, a point x ∈ Z such that Lcj∗xK• 6= 0. Therefore c ≤ d, butthis is absurd.

We recall the notion of Kodaira-Spencer map for families of sheaves. Letf : Z → S be a morphism of algebraic varieties and F a coherent sheaf on Z,flat over S, and let D = Spec k[ε]/ε2 be the double point scheme. The tangentspace TsS at a closed point s ∈ S is the space of the morphisms v : D → S

that map the closed point s0 of D to s. A tangent vector v ∈ TsS induces amorphism vD = (v× Id) : D×S Z → Z, which defines an infinitesimal deformationv∗DF of Fs. The set of such deformations is identified with the space Ext1

Zs(Fs,Fs)(cf. Proposition A.70 for a similar statement in derived category), and the Kodaira-Spencer map at the point s is the resulting morphism

KSs(F) : TsS → Ext1Zs(Fs,Fs) .

Assume now that A is a full subcategoy of the category of coherent sheaveson an algebraic variety X and that M is a functor associating to any variety T theset M(T ) of coherent sheaves E on T ×X flat over T such that Et is an object of A

for every (closed) point t ∈ T . As usual, two such families E and E ′ are consideredequivalent if E ' E ′⊗π∗1L, where L is a line bundle on T . We also assume that Mhas a fine moduli space, that is, it is represented by an algebraic variety M . Then,there is a universal sheaf Q on M ×X, flat over M inducing an equivalence

Hom(T,M) 'M(T )

φ 7→ [(φ× Id)∗(Q)] ,(1.19)


where square brackets denote equivalence clases. Now, Equation (1.19) gives anequivalence between TsM ' Hom(D,M)s and the infinitesimal deformations ofQs, that is, an isomorphism TsM ' Ext1

X(Qs,Qs), which coincides by its verydefinition with the Kodaira-Spencer map KSs(Q) for the universal family Q atthe point s. In other words, the Kodaira-Spencer map for the universal family Qat any point s ∈M is an isomorphism:

KSs(Q) : TsM ' Ext1X(Qs,Qs) . (1.20)

The following result is essentially contained in [61].

Lemma 1.23. Assume that the base field k has characteristic zero. Let S and X

be algebraic varieties, with X projective, and let F be a coherent sheaf on S ×X,flat over S and schematically supported on a closed subscheme j : Z → S × X.Suppose that for every closed point s ∈ S the support Z ∩ (s ×X) of the sheafFs is topologically a single point x ∈ X and that

HomX(Fs,Ox) ' k .

Then F is a line bundle on its support, i.e., F ' j∗L for a line bundle L on Z.Moreover, if for all pairs of distinct closed points s1, s2 ∈ S the sheaves Fs1 , Fs2are not isomorphic, then the Kodaira-Spencer map of the family F is injective atsome point s ∈ S.

Proof. Let us first prove that for every s ∈ S, Fs is the structure sheaf of a zero-dimensional closed subscheme of X, namely, that there is a surjective morphismf : OX → Fs; this will imply that Fs ' OZs . Let g : Fs → Ox and f : OX → Fsbe nonzero morphisms. If f is not surjective, coker f is a nonzero sheaf supportedtopologically at x and then there is a nonzero map coker f → Ox. This induces anonzero morphism h : Fs → Ox such that hf = 0. Now since HomX(Fs,Ox) ' k,the morphism g is a multiple of h and then g f = 0, i.e., f takes values in ker g.Since dimH0(X, ker g) = dimH0(X,Fs)− 1, there exist morphisms f : OX → Fsnot coming from H0(X, ker g) and these are surjective.

The remaining issues are local, so we may assume that S is affine. Let s ∈ Sbe a closed point. By hypothesis there exists a surjective morphism σs : OX → Fs.The natural morphism H0(S×X,F)→ H0(X,Fs) is surjective and therefore thereexists a global section σ of F mapping to σs. For a suitable open neighborhood Uof s, σU : OU → FU is still surjective, that is, FU is the structure sheaf of a closedsubscheme of U ×X. Then one must have FU ' OZU , where ZU = Z ∩ (U ×X),which proves that F is a line bundle on its support. Moreover, FU induces a mapα : U → HilbP (X) to the Hilbert scheme of zero-dimensional subschemes of Xwhose Hilbert polynomial P is equal to that of the sheaf Fs. The morphism α ischaracterized by the condition F ' (1 × α)∗Q, where Q is a universal sheaf on


HilbP (X) × X. By hypothesis the map α is injective on closed points, and thetangent map Tsα : TsU → Tα(s) HilbP (X) is injective at least at a point s ∈ U

(one here uses essentially the fact that k has characteristic zero). Moreover, theKodaira-Spencer map at s for the family FU is the composition of the tangentmap Tsα with the Kodaira-Spencer map for the universal family Q. Since thelatter is an isomorphism because of the universality of Q (cf. Eq. (1.20)), KSs(F)is injective.

Let X, Y be proper varieties and F a coherent sheaf on X × Y , flat over X.The flatness of F implies Fx ' ΦFX→Y (Ox) for every closed point x ∈ X.

Lemma 1.24. [61] The integral functor Φ = ΦFX→Y induces a morphism

Φ: Ext1X(Ox,Ox)→ Ext1

Y (Fx,Fx) (1.21)

which coincides with the Kodaira-Spencer morphism for the family F .

Proof. Note that in view of (1.8) one has the identification Ext1X(Ox,Ox) = TxX.

Given a tangent vector v ∈ TxX, the corresponding deformation of the sheafOx is the pullback v∗X(O∆) of the structure sheaf of the diagonal ∆ ⊂ X ×X. The morphism (1.21) sends v to the deformation of Φ(Ox) induced by it,namely, to the deformation ΦD(v∗X(O∆)). By base change (Proposition 1.8), onehas ΦD(v∗X(O∆)) ' v∗D(ΦX(O∆)) ' v∗D(F), where the last equality is due toProposition 1.10. But this is the image of v by the Kodaira-Spencer map of thefamily F .

1.3.2 Strongly simple objects

In this section the base field k has characteristic zero.

Let K• be a kernel in Db(X × Y ). By taking its restrictions to the fibers ofX × Y over X we obtain a family Lj∗xK• of objects in Db(Y ) parameterized bypoints in X. The notion of strong simplicity, originally stated for sheaves in [202],expresses a kind of orthonormality condition for the elements of this family.

Definition 1.25. A kernel K• in Db(X × Y ) is strongly simple over X if it satisfiesthe following two conditions:

1. HomiD(Y )(Lj

∗x1K•,Lj∗x2

K•) = 0 unless x1 = x2 and 0 ≤ i ≤ dimX;

2. Hom0D(Y )(Lj

∗xK•,Lj∗xK•) = k.

4


Since Lj∗xK• ' ΦK•

X→Y (Ox), one can check the conditions in this definition onthe groups Homi

D(Y )(ΦK•X→Y (Ox1),ΦK

•

X→Y (Ox2)).

Remark 1.26. For every pair of points x1, x2 in X there is an isomorphism

HomiD(Y )(Lj

∗x1K•∨,Lj∗x2

K•∨) ' HomiD(Y )(Lj

∗x2K•,Lj∗x1

K•) .

Therefore, a kernel K• in Db(X × Y ) is strongly simple over X if and only if itsdual K•∨ is so. 4

The following crucial result was originally proved by Bondal and Orlov [48].

Theorem 1.27. Let X and Y be smooth projective algebraic varieties, and let K•be a kernel in Db(X × Y ). The functor ΦK

•

X→Y is fully faithful if and only if K• isstrongly simple over X.

Proof. If the functor ΦK•

X→Y is fully faithful, one has

HomiD(Y )(Lj

∗x1K•,Lj∗x2

K•) ' HomiD(Y )(Φ

K•X→Y (Ox1),ΦK

•

X→Y (Ox2))

' HomiD(X)(Ox1 ,Ox2)

and then K• is strongly simple over X.

Let us assume in turn that K• is strongly simple over X. We need to provethat the functor ΦK

•

X→Y is fully faithful. We know from Proposition 1.3 that

ΦK•∨⊗π∗Y ωY [n]

Y→X ΦK•

X→Y ' ΦM•

X→X ,

where M• = Rπ13∗(π∗12K•L⊗π∗23(K•∨ ⊗ π∗Y ωY [n])) and πij denotes the projection

of X × Y ×X onto the (i, j)-th factor.

The strategy of the proof is as follows: first we prove thatM• is topologicallysupported on the image ∆ of the diagonal morphism δ : X → X × X. Next, weshow that M• is the push-forward of a line bundle N supported on a closedsubscheme Z of X × X. We then prove that Z coincides with the diagonal ∆.So the functor ΦM

•

X→X is the twist by N , as we saw in Example 1.2; this is anequivalence of categories and then ΦK

•

X→Y is fully faithful by Remark 1.21.

(a) M• ' M for a sheaf M topologically supported on the diagonal. Let usfix a point x1 ∈ X. For every point x2 ∈ X we have

(Lij∗x2ΦM

•

X→X(Ox1))∗ ' HomiD(X)(Φ

M•X→X(Ox1),Ox2)

' HomiD(Y )(Φ

K•X→Y (Ox1),ΦK

•

X→Y (Ox2)) ,

which is zero unless x1 = x2 and 0 ≤ i ≤ m by the strong simplicity of K•. Byapplying Lemma 1.22 to the immersion x1 → X one sees that ΦM

•

X→X(Ox1) '


Lj∗x1M• is a sheaf topologically supported at x1. By Proposition 1.11, M• ' M

whereM is a sheaf on X ×X flat over X for the first projection p1 : X ×X → X.Moreover, since j∗xM is topologically supported on x,M is topologically supportedon the diagonal.

(b) Let us denote by δ : Z → X × X the schematic support of M, so thatM = δ∗N for a coherent sheaf N on Z. We show that N is a line bundle. Forevery closed point x ∈ X the sheaf j∗xM ' ΦMX→X(Ox) is topologically supportedon x, and moreover one has

HomX(ΦMX→X(Ox),Ox) ' Hom0D(X)(Φ

K•X→Y (Ox),ΦK

•

X→Y (Ox) ' k (1.22)

because K• is strongly simple. So by Lemma 1.23, N is a line bundle.

(c) The scheme Z coincides with the diagonal ∆. A first step is to showthat the sheaf p1∗(M) is a line bundle on X. The diagonal embedding δ factorsthrough a closed immersion η : X → Z which topologically is a homeomorphism.The morphism p1 = p1 δ : Z → X being finite, the condition that M is flat overX through p1 implies that p1∗(M) ' p1∗(N ) is a locally free sheaf. Let r be itsrank. To see that r = 1, it is enough to prove that ΦMX→X(Ox) ' Ox for at leastone closed point x ∈ X. Let us consider the exact sequence

0→ Px → ΦMX→X(Ox)→ Ox → 0

where the last morphism is the adjunction and Px is the kernel. We want to provethat for some point x the sheaf Px is zero. Since Px is supported at x, it sufficesto see that HomX(Px,Ox) = 0. Taking homomorphisms in Ox, and in view ofEquation (1.22), we have an exact sequence

0→ HomX(Px,Ox)→ Hom1D(X)(Ox,Ox)→ Hom1

D(X)(ΦMX→X(Ox),Ox) ,

so that we have to show that Hom1D(X)(Ox,Ox) → Hom1

D(X)(ΦMX→X(Ox),Ox) is

injective. By Remark 1.21, we need only to prove that the morphism

ΦMX→X : Hom1D(X)(Ox,Ox)→ Hom1

D(X)(ΦMX→X(Ox),ΦMX→X(Ox))

is injective. Lemma 1.24 tells us that this morphism is the Kodaira-Spencer mapfor the family M, which is injective at some point x by Lemma 1.23.

Finally, we show that η : X → Z is an isomorphism, which is equivalent toprove that the finite morphism p1 : Z → X is an isomorphism. This follows fromthe fact that the direct image of the line bundle N is also a line bundle.

This characterization of fully faithful integral functors allows us to proverather easily that the product of two such functors is again fully faithful. Let X,Y and X, Y be smooth projective varieties, and let K•, K• be kernels in Db(X×Y )


and Db(X× Y ), respectively. These define a kernel K•L K• in Db(X×X×Y × Y ),

where K•L K• is the “box product” π∗X×YK• ⊗ π∗X×Y K

•.

Lemma 1.28. If the integral functors ΦK•

X→Y and ΦK•

X→Y are fully faithful, then the

functor ΦK•

L

K•X×X→Y×Y is fully faithful.

Proof. The Kunneth formula (see Theorem A.89)

HomhD(Y×Y )

(E•LF•,G•

LM•) '⊕

i+j=h

HomiD(Y )(E•,G•)⊗Homj

D(Y )(F•,M•)

implies that if K• is strongly simple over X and K• is strongly simple over X, then

K•L K• is strongly simple over X × X.

The definition of strongly simple object in Db(X × Y ) basically implies thatsuch an object is simple (thus justifying the teminology), since the restriction toevery fiber x × Y is simple. A more formal proof of this fact may be obtainedas follows.

Proposition 1.29. A strongly simple object K• in Db(X × Y ) is simple.

Proof. By Theorem 1.27 the functor Φ = ΦK•

X→Y is fully faithful. The explicitexpression for the right adjoint H of Φ (cf. Proposition 1.13) shows that it changesbase, that is, the base-changed functor HX is a right adoint to the functor ΦX =ΦK•X

X×X→X×Y : Db(X × X) → Db(X × Y ), so that the latter is fully faithful. SinceK• ' ΦX(O∆) by Proposition 1.10, we have

HomDb(X×Y )(K•,K•) ' HomDb(X×X)(O∆,O∆) ' k .

A different proof is given in [249, Lemma 1.12].

When the kernel is a coherent sheaf the notion of strong simplicity in Def-inition 1.25 looks more familiar. Assume that Q is a coherent sheaf on X × Y ,flat over X. Then, for any point x ∈ X, the derived category restriction Lj∗xQ ismerely the sheaf Qx = j∗xQ on x × Y and the two conditions in Definition 1.25are equivalent to the following:

1. HomiD(Y )(Qx1 ,Qx2) = 0 for every i ∈ Z whenever x1 6= x2;


2. Qx is simple for every x ∈ X, i.e., all its automorphisms are constant mul-tiples of the identity, HomOX (Q,Q) ' k.

Condition 1 is equivalent to condition 1 in Definition 1.25 because

HomiD(Y )(Qx1 ,Qx2) ' ExtiOX (Qx1 ,Qx2)

and since Y is smooth, we have HomiD(Y )(Qx1 ,Qx2) = 0 for any x1, x2 unless

0 ≤ i ≤ n = dimY .

Definition 1.30. If Q is a sheaf on X × Y , flat over X, which is strongly simple asa complex, we call it a strongly simple sheaf over X. 4

Remark 1.31. By Remark 1.26, the dual of a locally free strongly simple sheaf isstrongly simple. 4Example 1.32. If we take X = Y , the structure sheaf O∆ of the diagonal ∆ ⊂X ×X is strongly simple over both factors. 4

In the literature one usually finds a particular case of Theorem 1.27; this iswhat we shall mostly use in the sequel.

Theorem 1.33. A coherent sheaf Q on X × Y which is flat over X is stronglysimple over X if and only if the integral functor ΦQX→Y : Db(X) → Db(Y ) is fullyfaithful.

The following result is known as “Parseval formula” because it is similar tothe formula for the ordinary Fourier transform for functions on a torus bearingthe same name. We recall that if Φ = ΦK

•

X→Y is an integral functor and E is WITifor Φ, we denote by E = Φi(E) the unique nonzero Fourier-Mukai sheaf, and thatone has Φ(E) ' E [−i].

Proposition 1.34. Let Φ = ΦK•

X→Y a fully faithful integral functor.

1. One hasHomh

D(X)(F•, E•) ' HomhD(Y )(Φ(F•),Φ(E•))

for any pair of objects F• and E• of Db(X).

2. Let F a WITi sheaf and G a WITj sheaf on X. Then

ExthX(F ,G) ' Exth+i−jY (F , G) .

for every h ≥ 0. In particular, there is an isomorphism

ExthX(F ,F) ' ExthY (F , F)

for every h ≥ 0.


Proof. Since HomhD(X)(F•, E•) = HomD(X)(F•, E•[h]), the first statement holds

true because Φ is fully faithful. As for the second, we use the first formula togetherwith Φ(F) ' F [−i], Φ(E) ' E [−j] and Yoneda’s formula (cf. Proposition A.68).

1.4 The equivariant case

If an algebraic (typically, finite) group G acts on an algebraic variety X, one maydefine the equivariant derived category of coherent sheaves on X (cf. [39]). This isdefined in terms of coherent sheaves carrying a linearized action of G, compatiblewith the action on X. In this section we provide the basics of this theory, whichwe shall then use in Chapter 7, notably in connection with the proof of the McKaycorrespondence.

1.4.1 Equivariant and linearized derived categories

Let G be an algebraic group. We denote by µ : G×G→ G and e : Spec(k) → G

the product and the unity of G, respectively. If X is an algebraic variety, a leftaction of G on X is given by a morphism σ : G×X → X such that

• the following diagram is commutative

G×G×Xµ×IdX//

IdG×σ

G×X

σ

G×X σ // X

• the composition Spec(k)×X ' X e×IdX−−−−→ G×X σ−→ X is the identity.

If g ∈ G is a closed point, we denote also by g the compositionX ' g×X →G×X µ−→ X; it acts on closed points as g(x) = g · x = µ(g, x).

A G-linearization of a sheaf E of OX -modules is an isomorphism of sheaveson G×X

ψ : σ∗E ∼→ π∗2E ,

(where π2 : G × X → X is the projection) satisfying the cocycle condition (µ ×IdX)∗(ψ) = π∗23(ψ) (σ × IdG)∗(ψ).

We shall be particularly interested in the situation where G is a finite group.Then a linearization ψ of E , or better, the inverse isomorphism λ = ψ−1, is charac-terized by a family of isomorphisms λEg : E ∼→ g∗E fulfilling the conditions λE1 = Id

1.4. The equivariant case 25

and λEgh = h∗(λEg )λEh. By a G-linearized sheaf we understand a pair (E , λE) whereE is a G-equivariant sheaf and λE is (the inverse of) a G-linearization for E .Example 1.35. If X is a smooth variety acted on by a finite group G, the canonicalline bundle ωX is canonically linearized. The isomorphism g : X → X induces anisomorphism of sheaves g∗ωX ' ωX , and we take λωX as the inverse. 4Example 1.36. If a finite group G acts trivially on an algebraic variety X (i.e.,the action morphism σ : G×X → X is the projection π2), a G-linearization of Eis a representation G → AutOX (E), that is, is an action of G on E . We can thendenote by EG the subsheaf of G-invariant sections of E . 4

In the remainder of this section we assume that G is finite (see [39] for thegeneral case).

If (E , λE) and (F , λF ) are two linearized OX -modules, the local Hom sheafHomOX (E ,F) has a natural linearization defined by letting

λHomOX (E,F)g (φ) = λFg g∗(φ) (λEg )−1

on every invariant open subset U ⊆ X.

By taking global sections one obtains a right action of G on HomX(E ,F),given by φg = λ

HomOX (E,F)g (φ). One can then introduce the category ModG(X)

of G-linearized OX -modules by defining HomModG(X)((E , λE), (F , λF )) as the in-variant part HomG

X(E ,F) of HomX(E ,F). A simple computation shows that ifφ ∈ HomG

X(E ,F), the linearizations λEg induce linearizations on both ker(φ) andcoker(φ); using this fact one easily checks that ModG(X) is an Abelian category.One can also define the full Abelian subcategories QcohG(X) and CohG(X) ofG-linearized quasi-coherent and coherent sheaves.

The linearizations of E and F also endow the tensor product E ⊗ F witha natural linearization, so that the tensor product gives a functor QcohG(X) ×QcohG(X)→ QcohG(X).

Let ϕ : G → H be a morphism of finite groups. If Y is an algebraic varietyacted on by H, a morphism f : X → Y is ϕ-equivariant if it is compatible withthe actions of G and H, that is, if the following diagram is commutative:

G×Xϕ×f //

σ

H × Y

σ

X

f // Y .

When no confusion can arise, we shall simply say that f is equivariant.

If F is an H-linearized sheaf on Y , the inverse images f∗(λFh ) : f∗F →f∗(h∗F) ' h∗(f∗F) give an H-linearization for f∗F . By using ϕ we get a G-linearization for f∗. Therefore, the inverse image yields a right exact functor


f∗ : ModH(Y ) → ModG(X) which maps QcohH(Y ) and CohH(Y ) to QcohG(X)and CohG(X), respectively.

If we now take a G-linearized sheaf E on X, the direct images f∗(λEg ) : f∗E →f∗(g∗E) ' g∗(f∗E) give a G-linearization for f∗E . In order to induce an H-linearization we impose that ϕ is surjective. The kernel ker(ϕ) acts trivially on Y ,and one can consider the sheaf of ker(ϕ)-invariant sections (f∗E)ker(ϕ) (cf. Example1.36). The G-linearization of f∗(E) induces then an H-linearization of (f∗E)ker(ϕ).Then the direct image defines a left exact functor

ModG(X)fker(ϕ)∗−−−−→ModH(Y )

E 7→ fker(ϕ)∗ (E) = (f∗E)ker(ϕ) ,

which maps QcohG(X) to QcohH(Y ). We call it the equivariant direct image. Ifin addition f is proper, the equivariant direct image f

ker(ϕ)∗ maps CohG(X) to

CohH(Y ).

Example 1.37. Let G be a finite group acting freely on a projective smooth vari-ety X. Then there is a geometric quotient Y = X/G and it is also smooth andprojective. The quotient morphism φ : X → Y is equivariant (with respect to thenatural projection f : G→ 1) and one easily checks that the functors

π∗ : Coh(Y )→ CohG(X) , πG∗ : CohG(X)→ Coh(Y )

are quasi-inverse of each other and hence, equivalences of categories (cf. [229]). 4

By a classical result of Grothendieck [133, Prop. 5.1.2], the category QcohG(X)has enough injectives. One can then take G-linearized injective resolutions to rightderive any left-exact functor. Examples are the global G-invariant homomorphismsF 7→ HomG

X(E ,F) and the equivariant direct image fker(ϕ)∗ for an equivariant map

f (when ϕ is surjective). The derived functors of the G-invariant homomorphismsare usually denoted ExtG,iX (E ,F), and they are naturally isomorphic to the G-invariant parts of the ordinary ExtiX(E ,F) for its natural G-action.

If we assume that X is projective, since G is finite there exists a G-equivariantample line bundle, and this implies that every G-invariant sheaf has a left reso-lution by locally free G-equivariant sheaves. Thus, we can left derive any rightexact functor on QcohG(X). In the case of the functor HomG we obtain the G-invariant Ext’s as defined before. We can also left-derive the tensor product ⊗ andthe inverse image f∗ under an equivariant map.

The G-linearized categories, being Abelian, have associated derived cate-gories. In particular, we can consider the full subcategory DG(X) of the de-rived category D(QcohG(X)) consisting of complexes with coherent cohomologysheaves, and the corresponding subcategories DG,b(X), DG,+(X) and DG,−(X) of


bounded, bounded below, and bounded above complexes, respectively. The nat-ural functor D(CohG(X)) → D(QcohG(X)) induces an equivalence between thebounded derived categories Db(CohG(X)) and DG,b(X).

By proceeding as in the case of usual derived categories of coherent sheaves,

we can introduce theG-equivariant derived functors E•L⊗F• and RHomOX (E•,F•)

for complexes E• and F• in DG,b(X). Much in the same way as in Corollary A.88one shows that if E• is an object in DG,b(X) of finite homological dimension, then

the functor (−)L⊗E•∨ : DG,b(X) → DG,b(X) is both left and right adjoint to the

functor (−)L⊗E• : DG,b(X)→ DG,b(X); namely, there are functorial isomorphisms

HomDG,b(X)(F•,G•L⊗E•) ' HomDG,b(X)(F•

L⊗E•∨,G•)

HomDG,b(X)(F•,G•L⊗E•∨) ' HomDG,b(X)(F•

L⊗E•,G•)

(1.23)

for F• and G• in DG,b(X).

Let us fix in the rest of this section a surjective morphism of groups ϕ : G→H, two algebraic varieties X, Y acted on by G and H respectively, and a properequivariant morphism f : X → Y . We then have the derived functors

Rfker(ϕ)∗ : DG,b(X)→ DH,b(Y ) , Lf∗ : DH,b(Y )→ DG,b(X) .

Both functors are compatible with composition of equivariant morphisms. Letψ : H → K be another group morphism and f : Y → Z a ψ-equivariant morphismof algebraic varieties (where K acts on Z); then, for every object F• in DK,−(Z)there is a functorial isomorphism Lf∗(Lf∗F•) ' L(f f)∗(F•) in DG,−(X). Forthe direct image, we have to assume that ψ is surjective. Then, for every objectE• ∈ DG(X), there is a functorial isomorphism

Rfker(ψ)∗ (Rfker(ϕ)

∗ (E•)) ' R(f f)ker(ψϕ)∗ (E•) . (1.24)

in DK(Z).

As in the ordinary case, Rfker(ϕ)∗ is a right adjoint to Lf∗, that is, there is a

functorial isomorphism

HomDG,b(X)(Lf∗F•, E•) ' HomDH,b(Y )(F•,Rf

ker(ϕ)∗ E•) . (1.25)

One also has an equivariant projection formula in DH,b(Y )

Rfker(ϕ)∗ (E•)

L⊗F• ' Rfker(ϕ)

∗ (F•L⊗Lf∗F•) , (1.26)

for E• in DG,b(X) and F• in DH,b(Y ).


Example 1.38. Let G be a finite group acting freely on a projective smooth varietyX and φ : X → Y quotient morphism. A direct consequence of Example 1.37 isthat the functors

Lπ∗ : Db(Coh(Y ))→ Db(CohG(X)) , RπG∗ : Db(CohG(X))→ Db(Coh(Y )) ,

are quasi-inverse of each other and hence, equivalences of categories. Moreover,they induce equivalences of categories

Lπ∗ : Db(Y ) ' DG,b(X) , RπG∗ : DG,b(X) ' Db(Y ) ,

which are also inverse of each other. An analogous statement is true for the derivedcategories of bounded below, bounded above and unbounded complexes. 4

The last standard property we would like to mention is the equivariant flatbase change. If ψ : K → H is another group morphism, Z is an algebraic varietyacted on by K and φ : Z → Y is a ψ-equivariant morphism, we can considerdiagrams

K ×H Gψ //

ϕ

G

ϕ

K

ψ // H

Z ×Y XφX //

fZ

X

f

Z

φ // Y

of group morphisms and of morphisms of algebraic varieties, respectively. Eachmorphism in the second diagram is equivariant with respect to the correspondingmorphism in the first diagram. Note that ϕ is still surjective and ker ϕ ' kerϕ.Then an equivariant analogue to Proposition A.85 holds true, namely, if either for φ is flat, there is a functorial isomorphism in DK,b(Z)

φ∗Rfker(ϕ)∗ E• ' Rfker(ϕ)

Z∗ (φ∗XE•) (1.27)

for any complex E• in DG,b(X).

A more delicate issue concerns Grothendieck duality for a proper equivariantmorphism f : X → Y . Luckily enough, the general duality formalism developed in[234] also applies to this case, once one checks that the equivariant direct imagefunctor is compatible with small coproducts; this can be seen as in the case of theordinary direct image. Then, as a consequence of the results in [234], the equiv-ariant derived direct image functor Rfker(ϕ)

∗ : DG,b(X) → DH,b(Y ) has a rightadjoint fker(ϕ),! : DH,b(Y )→ DG,b(X), that is, there is a functorial isomorphism

HomDH,b(Y )(Rfker(ϕ)X∗ F•,G•) ' HomDG,b(X)(F•, fker(ϕ),!G•) . (1.28)

The complex fker(ϕ),!OY is called the G-linearized dualizing complex of f , andwhenever f is smooth of relative dimension n, one has f !,ker(ϕ)OY ' ωX/Y [−n]in DG,b(X), where we endow the ordinary relative dualizing sheaf ωX/Y with itsnatural linearization, which is defined as in Example 1.35.


1.4.2 Equivariant integral functors

Since we have defined direct and inverse images and tensor product for linearizedcomplexes, one can define integral functors in the equivariant setting. Let G, H befinite groups acting on two proper algebraic varieties X and Y , respectively. ThenG×H acts naturally on the product X×Y , and the projections πX : X×Y → X,πY : X × Y → Y are equivariant with respect to the surjective group morphismsφG : G ×H → G and φH : G ×H → H, respectively. Note that kerφG ' H andkerφH ' G.

We can now consider “linearized kernels,” that is, objects K• in the linearizedderived category DG×H,−(X × Y ). The equivariant integral functor with kernelK• is the functor ΦK

•,G×HX→Y : DG,−(X)→ DH,−(Y ) given by

ΦK•,G×H

X→Y (E•) = RπGY ∗(π∗XE•

L⊗K•) , (1.29)

where all the functors involved are taken in the linearized sense.

Most of the results about integral functors previously described apply toequivariant integral functors, due to the properties described in Section 1.4.1. Wereport here a few properties that will be relevant to the proof of the derived McKaycorrespondence, which we shall discuss in Chapter 7.

When K• is of finite Tor-dimension as a complex of OX -modules, one canproceed as in the proof of Proposition 1.4 to prove that ΦK

•,G×HX→Y is bounded and

can be extended to a functor ΦK•,G×H

X→Y : DG(X) → DH(Y ) which maps DG,b(X)to DH,b(Y ). As for ordinary integral functors, the composition of two linearized in-tegral functors is obtained by convoluting in the linearized sense the correspondingkernels; if Z is another algebraic variety acted on by a finite group K, given twokernels K• in DG×H,−(X×Y ) and L• in DH×K,−(Y ×Z) corresponding to equiv-ariant integral functors Φ and Ψ, the composition Ψ Φ: DG,−(X) → DK,−(Z)has kernel

L• ∗ K• = RπHXZ∗(π∗XYK•

L⊗π∗Y ZL•)

in DG×K,−(X×Z), where the morphisms πXY , πXZ∗ and πY Z are the projectionsof the product X × Y ×Z onto the fiber products X × Y , X ×Z and Y ×Z, andall functors are taken in the linearized sense. The proof is analogous to that ofProposition 1.3, and uses Equation (1.24), together with the linearized flat basechange formula (1.27) and the linearized projection formula (1.26).

Adjoints to equivariant integral functors can be computed as for ordinaryintegral functors (cf. Proposition 1.13), using the properties of Grothendieck dual-ity in the linearized setting, and in particular Equation 1.28. For future referencelet us describe the adjunction property we shall need. We consider a linearizedkernel K• in DG×H,b(X×Y ) of finite Tor-dimension as a complex of OX -modules,and the corresponding linearized integral functor. Using the adjunction properties


given by Equations (1.23), (1.25) and (1.28), and proceeding as in the proof ofProposition 1.13, we obtain the following description of the adjoint of a linearizedintegral functor.

Proposition 1.39. Assume that K• is of finite Tor-dimension as a complex ofOX-modules and that Y is smooth. The kernel K•∨ ⊗ π∗Y ωY [n] is an object ofDG×H,b(X ×Y ) of finite Tor-dimension, and the corresponding linearized integralfunctor

ΦK•∨⊗π∗Y ωY [n],H×G

Y→X : DH,b(Y )→ DG,b(X) ,

where n = dimY , is a left adjoint to ΦK•,G×H

X→Y .

1.5 Notes and further reading

The first systematic investigation of integral functors is contained in Mukai’s sem-inal paper [224], where X is an Abelian variety, Y its dual variety and the kernelin Db(X × Y ) is provided by the normalized Poincare bundle (cf. Chapter 3).Subsequently, some authors started studying integral functors in more generality,notably Maciocia [202] and Bondal-Orlov [48, 49]. In particular, in [202] the no-tion of strong simplicity was first introduced, although it had already been usedimplicitly in [48, 50]. The theory has been somehow settled by Bridgeland’s paper[61]. Serre functors have been studied to some extent by Bondal and Kapranov[45].

The characterization of fully faithful integral functors in terms of the kernel(cf. Theorem 1.27) has been generalized to Gorenstein varieties [144] and Cohen-Macaulay varieties [143].

Equivariant integral functors and autoequivalences of equivariant derived cat-egories have been studied by Ploog in [250].

Chapter 2

Fourier-Mukai functors

Introduction

According to a fundamental theorem due to D. Orlov, any equivalence betweenderived categories of coherent sheaves of two smooth projective varieties is an inte-gral functor. This “representability” result — which lies at the heart of the currentchapter — opens the way to the investigation of the geometric consequences ofthe equivalence between the derived categories of two varieties.

Section 2.1 is quite technical; it presents and develops the notions of span-ning class, ample sequence and convolution (of a complex of objects in the derivedcategory). These will be mainly used in the next section and may at first be givenlittle attention, concentrating on definitions and main results, leaving proofs anddetails to a second reading. Section 2.2 pivots around Orlov’s theorem. An im-portant ingredient of the proof that we provide for this result is a construction ofthe resolution of the diagonal of a projective variety which generalizes Beılinson’sresolution of the diagonal of projective space. This generalization is originally dueto Kapustin, Kuznetsov and Orlov and has been further formalized by Kawamata.In Section 2.3 we specialize our attention to those integral functors which estab-lish an equivalence of categories. We call such functors Fourier-Mukai functors,reserving the term Fourier-Mukai transforms to the cases where the associatedkernel is a concentrated complex (i.e., it is a sheaf). One of the main objectivesof the section is to state and prove (basically following [61]) a criterion for testingwhether a fully faithful integral functor is a Fourier-Mukai functor. The existenceof an equivalence between the derived categories of two varieties of course severelyconstrains their geometry, and indeed one proves that whenever two smooth pro-

canonical bundle, then they are isomorphic. The section also provides other geo-

Progress in Mathematics 276, DOI: 10.1007/b11801_2,31C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics,

jective varieties have equivalent derived categories, and one of them has an ample


32 Chapter 2. Fourier-Mukai functors

metric applications, some of them concerning moduli spaces of sheaves.

2.1 Spanning classes and equivalences

We introduce the notion (due to Bridgeland [61] but already implicit in [48]) ofspanning class for a triangulated category.

Definition 2.1. Let A be a triangulated category. A subclass Σ ⊂ Ob(A) is aspanning class if the following two properties are satisfied:

1. if HomiA(σ, a) = 0 for all σ ∈ Σ and all i ∈ Z, then a = 0;

2. if HomiA(b, σ) = 0 for all σ ∈ Σ and all i ∈ Z, then b = 0.

4

Example 2.2. The skyscraper sheaves Ox form a spanning class for the derivedcategory Db(X) of a smooth projective variety, as we shall see in Proposition 2.52.Moreover, if L is an ample sheaf, then Lii∈Z is also a spanning class for Db(X)by virtue of Proposition 2.9. In particular OPn(i)i∈Z is a spanning class for thebounded derived category of the projective n-space. (Actually, for any m ∈ Z thecollection Lii<m is a spanning class as well.) 4

Definition 2.3. A triangulated category A is decomposable if there exist two trian-gulated nontrivial full subcategories A1, A2 such that

1. For every object a in A there exist objects ai in Ai and a triangle

a1 → a→ a20−→ a1[1] ;

2. For every pair of objects a1, a2 in A1,A2 one has

HomiA(a1, a2) = Homi

A(a2, a1) = 0 , for any i ∈ Z.

We then write A ' A1 ⊕ A2. A triangulated category A is indecomposable if it isnot decomposable. 4

Lemma 2.4. Let F : A → B be an exact fully faithful functor with a right adjointH and a left adjoint G. Assume that B is indecomposable and that A is nontrivial.Then F is an equivalence if and only if the condition H(c) = 0 for an object c inB implies G(c) = 0.

2.1. Spanning classes and equivalences 33

Proof. If F is an equivalence, then G ' H, and there is nothing to prove. For theconverse, define full subcategories B1, B2 consisting of the objects b in B suchthat FH(b) ' b and H(b) = 0 respectively. If b1, b2 are objects in B1,B2, one has

HomiB(b1, b2) ' Homi

B(FH(b1), b2) ' HomiA(H(b1), H(b2)) = 0

andHomi

B(b2, b1) ' HomiB(b2, FH(b1)) ' Homi

A(G(b2), H(b1)) = 0

because G(b2) = 0 by hypothesis. Moreover, for any object b in B the counitmorphism ε(b) : FH(b)→ b can be embedded into a triangle

FH(b)ε(b)−−→ b→ c

δ−→ FH(b)[1] .

Applying H we get H(c) = 0; in fact, H(ε(b)) : HFH(b)→ H(b) is an isomorphismsince F is fully faithful. Then FH(b) is an object of B1 and c is in B2. Moreover the

composition of 0 = H(c)H(δ)−−−→ HFH(b)[1] ' Hb[1] is the morphism corresponding

by adjunction to c δ−→ FH(b)[1], and then δ = 0. Since B is indecomposable, eitherB1 or B2 is trivial. If B1 is trivial, then B2 = B so that H = 0; then for everyobject a of A we have HomA(a, a) ' HomB(F (a), F (a)) ' HomA(a,H(F (a))) = 0and this implies that A is trivial. But this is impossible, and therefore B2 istrivial, which means that c = 0 and ε(b) : FH(b) ∼→ b for any object b in B. ThusFH ' IdB, and F is an equivalence.

As shown in Corollary 1.18, exact equivalences of categories intertwine Serrefunctors. Whenever this intertwining property holds true for all objects in a span-ning class, under some additional assumptions one has a converse statement.

Proposition 2.5. Let F : A → B be an exact fully faithful functor of triangulatedcategories with Serre functors SA, SB. Assume that B is indecomposable, A is nottrivial and that F has a right adjoint H. Then F is an equivalence if and only ifSBF (σ) = FSA(σ) for all σ in some spanning class Σ ⊂ Ob(A).

Proof. By Lemma 1.17, F has a left adjoint given by G = S−1A H SB. For any

object b in B, any σ ∈ Σ and any i ∈ Z, we have

HomiA(σ,G(b)) ' Homi

A(G(b), SA(σ))∗ ' HomiA(b, FSA(σ))∗

' HomiA(b, SBF (σ))∗ ' Homi

A(F (σ), b) ' HomiA(σ), H(b)) .

Then G(b) = 0 if and only if H(b) = 0, so that F is an equivalence by Lemma2.4.

Spanning classes may be used to test whether a functor is fully faithful.


Theorem 2.6. [61] Let F : A → B be an exact functor of triangulated categories,admitting a left and a right adjoint. The functor F is fully faithful if and only ifthe morphism

F : HomiA(σ, τ)→ Homi

B(F (σ), F (τ))

is an isomorphism for every σ, τ in some spanning class Σ for A.

Proof. Let H,G be a right and a left adjoint to F and consider the correspondingunits and counits

η : IdA → H F ε : F H → IdB

ξ : IdB → F G δ : G F → IdA

We have a commutative diagram

HomiA(σ, τ)

η(τ)∗ //

δ(σ)∗

F

))SSSSSSSSSSSSSSSHomA(σ,H F (τ))

β '

HomiA(G F (σ), τ) α

' // HomiB(F (σ), F (τ))

(2.1)

where α = ξ(F (σ))∗ F and β = ε(F (τ))∗ F are the isomorphisms given byadjunction. Since F is an isomorphism for all σ, τ ∈ Σ, all morphisms in diagram2.1 are isomorphisms for all σ, τ ∈ Σ. Taking this into account we proceed in threesteps:

(1) We prove that δ(σ) : G F (σ)→ σ is an isomorphism for every σ ∈ Σ.

Indeed, δ(σ) fits into the triangle G F (σ)δ(σ)−−−→ σ → ρ→ G F (σ)[1]. For every

τ ∈ A we have an exact sequence

· · · → Hom−1A (σ, τ)

δ(σ)∗−−−→ Hom−1A (G F (σ), τ)→

HomA(ρ, τ)→ HomA(σ, τ)δ(σ)∗−−−→ HomA(G F (σ), τ)→ . . .

If τ ∈ Σ, then all morphisms δ(σ)∗ in this exact sequence are isomorphisms, sothat Homi

A(ρ, τ) = 0 for every i ∈ Z and ρ = 0. Thus δ(σ) is an isomorphism.

(2) Next we show that η(τ) : τ → H F (τ) is an isomorphism for every

τ ∈ Ob(A). We can embed η(τ) into the triangle τη(τ)−−−→ H F (τ)→ ρ→ τ [1] and

get for every σ ∈ Σ the exact sequence

· · · → HomA(σ, τ)η(τ)∗−−−→ HomA(σ,H F (τ))→ HomA(σ, ρ)→

Hom1A(σ, τ)

η(τ)∗−−−→ Hom1A(σ,H F (τ))→ . . .


As σ ∈ Σ, by (1) δ(σ) is an isomorphism, hence δ(σ)∗ is an isomorphism. Bydiagram (2.1) F is an isomorphism, so that η(τ)∗ is an isomorphism as well forevery σ ∈ Σ, τ ∈ Ob(A). By the above exact sequence one has Homi

A(σ, ρ) = 0 forevery i ∈ Z, σ ∈ Σ and thus ρ = 0 and η(τ) is an isomorphism.

(3) Finally, we prove that F is fully faithful. Since η(τ) is an isomorphismfor every τ ∈ Ob(A), η(τ)∗ is an isomorphism for every σ, τ ∈ Ob(A), and then Fis an isomorphism by diagram (2.1).

2.1.1 Ample sequences

Let A be a k-linear Abelian category.

Definition 2.7. A sequence Pii∈Z of objects in A is said to be ample if for everyobject C of A there is an integer i0 = i0(C) such that the following conditionshold for i < i0:

1. the natural morphism HomA(Pi, C)⊗ Pi → C is surjective;

2. HomjDb(A)

(Pi, C) = 0 for j 6= 0;

3. HomA(C,Pi) = 0.

4

Condition 1 is equivalent to the existence of an exact sequence

P⊕sjα−→ P⊕ki

ρ−→ A→ 0 .

The most important example is provided by the sequence Lii∈Z in thecategory of quasi-coherent sheaves on a projective variety where L is an ampleline bundle (here Pi = L−i).

The following results are used in the proof of Orlov’s representability theorem2.15.

Lemma 2.8. [242] Let Pii∈Z be an ample sequence in A. An object A• of Db(A)is isomorphic to an object of A (i.e., it is isomorphic in Db(A) to a complexconcentrated in degree zero) if and only if Homj

Db(A)(Pi,A•) = 0 for j 6= 0 and

i 0.

Proof. The “only if” part is clear by the definition of ample sequence. Now assumethat Homj

Db(A)(Pi,A•) = 0 for j 6= 0 and i 0. Embed A into an Abelian category

with enough injectives, so that HomjDb(A)

(Pi, C) = Extj(Pi, C), where the Extsare computed in the larger category. Since A• has bounded cohomology, we can


find i 0 such that Extj(Pi, Hq(A•)) = 0 for every q. Then there is a convergentspectral sequence Ep,q2 = Extp(Pi, Hq(A•)) with Ep+q∞ = Homp+q

Db(A)(Pi,A•). If

Hq(A•) 6= 0, we have E0,q2 = Hom(Pi, Hq(A•)) 6= 0 for i 0 by the first condition

in Definition 2.7, so that any nonzero element in E0,q2 survives to infinity yielding

a nonzero element in Eq∞ = HomqDb(A)

(Pi,A•). Thus q = 0 and A• ' H0(A•) inDb(A).

Proposition 2.9. Let Pii∈Z be an ample sequence in A. Assume that Db(A) hasa Serre functor. Then the class Σ = Pii∈Z ⊂ Ob(Db(A)) is a spanning class forDb(A).

Proof. Let A• be an object of Db(A) and assume that HomjDb(A)

(Pi,A•) = 0for every i and j. By Lemma 2.8, A• is isomorphic to an object A in A. Bythe first condition in Definition 2.7, one has A = 0. Now take a complex A• inDb(A) such that Homj

Db(A)(A•, Pi) = 0 for every i and j. If S is the Serre functor

of the category Db(A), then HomjDb(A)

(Pi, S(A•)) ' HomDb(A)(Pi, S(A•[j])) 'HomDb(A)(A•[j], Pi)∗ ' Hom−j

Db(A)(A•, Pi)∗ = 0 for every i and j. Then by the

previous argument we have S(A•[j]) = 0 for every j, so that A• = 0.

We shall also denote by Σ the full subcategory of Db(A) whose objects arePii∈Z.

Proposition 2.10. [242, Prop. 2.16] Let Pii∈Z be an ample sequence of objects inA. Let F : Db(A)→ Db(A) be an exact equivalence. Every isomorphism h : IdΣ

∼→F|Σ can be extended to an isomorphism IdDb(A)

∼→ F on Db(A).

Proof. We have for every i an isomorphism hPi : Pi ∼→ F (Pi) depending functori-ally on Pi. Our task is to extend these isomorphisms to functorial isomorphismshA• : A• → F (A•) for every object A• in Db(A). We divide this rather long proofinto four steps.

(1) A• ' ⊕nk=1Pik . In this case F (A•) ' ⊕nk=1Pik and we simply set hA• to bethe direct sum of the given hPik .

(2) A• is isomorphic to an object A of A. We first note that F (A) is an object ofA since

HomjDb(A)

(Pi, F (A)) ' HomjDb(A)

(F (Pi), F (A)) ' HomjDb(A)

(Pi, A)

due to Pi ' F (Pi), and we can apply Lemma 2.8. By definition of ample sequencethere is an exact sequence

P⊕sjα−→ P⊕ki

ρ−→ A→ 0


for i, j 0. We then have a diagram

P⊕sjα //

hsPj'

P⊕kiρ //

hkPi'

A // 0

F (P⊕sj )F (α) // F (Pi)⊕k

F (ρ) // F (A) .

Since F (ρ) hkPi α = F (ρ) F (α) hsPj = F (ρ α) hsPj = 0, there is an iso-morphism hA : A ∼→ F (A) which completes the diagram. Moreover hA is uniqueas Hom(A,F (A)) → Hom(F (Pi)⊕k, F (A)). This also implies that hA is indepen-dent of the choice of the surjection P⊕ki

ρ−→ A→ 0. One can easily check that hAdepends functorially on A.

(3) A• ' A[n] for an object A of A. Since F (A[n]) ' F (A)[n] we simply sethA[n] = hA[n].

(4) A• is any object in Db(A). We proceed by induction on the length N =`(A•), which is the number of nonzero cohomology objects of A•. The case N = 1corresponds to the objects of A and has been considered in the previous steps. Letus then take N > 1.

Let q be the maximum of the integers such that Hq(A•) 6= 0. Then we canfind an index i and a surjective morphism P⊕ki → ker dq inducing a morphismφ : P⊕ki [−q] → A•≤q ' A• in the derived category. We can also choose the indexi so that:

(a) the induced morphism Hq(φ) : P⊕ki → Hq(A•) is surjective;

(b) HomjDb(A)

(Pi, Hp(A•)) = 0 for all j 6= 0 and for all p;

(c) HomjDb(A)

(Hq(A•), Pi) = 0 for all j 6= 0 and for all q.

We can now embed φ into an exact triangle B• → P⊕ki [−q] φ−→ A• → B•[1]. Since`(B•) = N − 1, induction provides a commutative diagram

B•[−1] //

hB• [−1]'

P⊕ki [−q]φ //

hP⊕ki

[−q]'

A•ψ // B•

hB•'

F (B•)[−1] // F (P⊕ki )[−q]F (φ) // F (A•)

F (ψ) // F (B•) .

Then there exists an isomorphism hA• : A• ∼→ F (A•) making the above diagraminto a morphism of triangles. Moreover, this morphism is the only one satisfyingthe condition F (ψ) hA• = hB• ψ, as

HomDb(A)(A•, F (P⊕ki )[−q]) ' HomDb(A)(A•, P⊕ki [−q])' Homq

Db(A)(A•, P⊕ki ) = 0


Again, one easily proves that the morphism hA• does not depend on thechoice of the morphism φ : P⊕ki [−q]→ A•≤q ' A•.

Finally, we prove that this construction is functorial. Let us take a morphism$ : A• → C• with `(C•) ≤ N . We must prove that the diagram

A• $ //

hA•'

C•

hC•'

F (A•)F ($) // F (C•)

(2.2)

is commutative. Let q be as above the maximum of the integers such thatHq(A•) 6=0 and p the maximum of the integers such that Hp(C•) 6= 0. We consider separatelythe cases p < q and p ≥ q.

The case p < q. We take as above a morphism φ : P⊕ki → A• and thecorresponding exact triangle

P⊗ki [−q] φ−→ A• ψ−→ B• → P⊗ki [−q + 1] .

We can assume HomDb(A)(Pi, Hj(C•)) = 0 for all j, so that HomDb(A)(Pi, C•) = 0.Then $ factors through a morphism ρ : B• → C•. Since `(B•) = `(A•) − 1, thediagram

B•ρ //

hB•'

C•

hC•'

F (B•)F (ρ) // F (C•)

commutes by induction. Then (2.2) is commutative as well.

The case p ≥ q. We proceed by induction on p − q, the case p − q = −1being included in the preceding case. We can find an index i and a surjectivemorphism P⊕ki → ker dp. This gives a morphism φ : P⊕ki [−p] → C•≤p ' C• inthe derived category. Moreover, the induced morphism Hp(φ) : P⊕ki → Hp(C•) issurjective. We can also assume that HomDb(A)(Hj(A•), P⊕ki ) = 0 for all j, so thatHomDb(A)(A•, P⊕ki [−p]) = 0. We have an exact triangle

P⊕ki [−p] φ−→ C• β−→M• → P⊕ki [−p+ 1]

and the maximum of the integers p with H p(M•) 6= 0 is smaller than p. Let us


consider the diagram

C•

hC•'

β

%%LLLLLLLLLL

A•β$ //

hA•'

$

99ttttttttttM•

hM•'

F (C•)F (β)

%%JJJJJJJJJ

F (A•)F (β$) //

F ($)::vvvvvvvvv

F (M•) .

By induction, the rectangular-shaped subdiagram is commutative, and by theconstruction of the morphisms hC• the lozenge-shaped subdiagram on the rightis commutative as well. Then F (β) (hC• $ − F ($) hA•) = 0 and sinceHomDb(A)(Hp(A•), P⊕ki ) = 0 we obtain that hC• $ − F ($) hA• = 0, thatis, the diagram (2.2) is commutative.

Proposition 2.11. [176, Lemma 6.5] Let F, F : Db(A) → B, be exact functors,where B is a k-linear triangulated category. Assume that:

1. Db(A) has a Serre functor;

2. F, F have left adjoint functors G and G and right adjoint functors H, H(note that by Lemma 1.17, if B has a Serre functor, then right adjoints ifand only if left adjoints exist);

3. F is fully faithful;

4. there is an ample sequence Pii∈Z of objects of A and an isomorphism offunctors fΣ : F|Σ ∼→ F|Σ, where Σ is the full subcategory of Db(A) whoseobjects are Pii∈Z.

Then there is an isomorphism of functors f : F ∼→ F extending fΣ.

Proof. Since the restrictions of F and F to Σ are isomorphic and Pii∈Z is aspanning class by Proposition 2.9, Theorem 2.6 implies that F is fully faithful.Since F and F are fully faithful, there are functor isomorphisms IdDb(A)

∼→ HF

and GF ∼→ IdDb(A).

Now, GF is left adjoint to HF . Since both composed functors are isomorphicto the identity when rectricted to Σ, again by Theorem 2.6 they are fully faithful.Whenever a fully faithful functor admits a left adjoint which is fully faithful, then itis an equivalence of categories. Thus, HF is an exact equivalence. By Proposition


2.10, the isomorphism IdΣ∼→ HF |Σ can be extended to a functor isomorphism

IdDb(A)∼→ HF . Composing from the left with F and using adjunction, we have a

morphism f : F → F extending fΣ.

Let us check that f is an isomorphism. For any object A• in Db(A), let

F (A•) fA•−−→ F (A•)→ c→ F (A•)[1]

be an exact triangle. Since H(fA•) is an isomorphism we have H(c) = 0 and then

HomDb(A)(Pi, H(c)) ' HomB(F (Pi), c)

' HomB(F (Pi), c) ' HomDb(A)(Pi, H(c)) = 0

for every i, so that H(c) = 0. Thus HomB(F (A•), c) = 0, and F (A•) ' F (A•)⊕ cfrom the above exact triangle. Since HomB(F (A•), c) ' HomDb(A)(A•, H(c)) = 0we deduce that c = 0 and then fA• is an isomorphism.

2.1.2 Convolutions

Given an object in Abelian category, we are used to associate with it a complexof objects in the derived category; we also know how to associate an object ofthe derived category with a double complex. Sometimes, dealing with (bounded)complexes of objects in the derived category

(A•)−m d−m−−−→ (A•)−m+1 d−m+1−−−−→ · · · → (A•)−1 d−1−−→ (A•)0 ,

one wishes to construct objects a of the derived category which somehow representthem; one also requires that when the objects of the complex we start with are inthe original Abelian category,

A• ≡ A−m d−m−−−→ A−m+1 d−m+1−−−−→ · · · → A−1 d−1−−→ A0

the new object a is just the image A• of the complex in the derived category.

This is possible under very mild requirements, and the process is called con-volution. Let us consider at first a complex

A• ≡ A−1 d−1−−→ A0

of objects of A. If Cone(d−1) is the cone of d−1, the natural morphism A0 →Cone(d−1) induces an isomorphism in the derived category

A• ∼→ Cone(d−1) .


Then Cone(d−1) represents the complex A• in the derived category Db(A). Thisis definitely a trivial observation, but this “cone construction” may be straightfor-wardly extended to complexes of objects in the derived category: given a complex

(A•)−1 d−1−−→ (A•)0

of objects of Db(A), we can take a cone Cone(d−1) of d−1 and we have a naturalmorphism (A•)0 → Cone(d−1) and an exact triangle

(A•)−1 d−1−−→ (A•)0 → Cone(d−1)→ (A•)−1[1] .

We can iterate this process to define the right convolution of a bounded complexof objects of the derived category. Actually, we do not need to work with a derivedcategory, since any triangulated category will do.

Let then B be a triangulated category and

a−md−m−−−→ a−(m−1) d−(m−1)−−−−−→ a−(m−2) → · · · → a−1 d−1−−→ a0 (2.3)

a complex of objects of B (that is, the composition of any two consecutive mor-phisms vanishes). Assume also that one has

HomB(a−p[r], a−q) = 0 , for every p > q and r > 0. (2.4)

Following Orlov and Kawamata [242, 176] we can define the right convolutionof the complex 2.3 as the pair formed by the object a of B and the morphismd0 : a0 → a constructed by induction on the length m as follows:

• If m = 0, then a = a0 and d0 is the identity.

• If m ≥ 1, we let a−(m−1) = Cone(d−m), so that there is an exact triangle

a−md−m−−−→ a−(m−1) g−(m−1)−−−−−→ a−(m−1) → a−m[1] .

After taking homomorphisms we have an exact sequence

HomB(a−m[1], a−(m−2))→ HomB(a−(m−1), a−(m−2))→

HomB(a−(m−1), a−(m−2))→ HomB(a−m, a−(m−2)) .

Since dm−2 dm−1 = 0 there is a morphism dm−1 : a−(m−1) → a−(m−2) such thatdm−1 gm−1 = dm−1; due to condition (2.4) one also has HomB(a−m[1], a−(m−2))= 0 and then the morphism is unique. Hence, we obtain a new complex

a−(m−1) d−(m−1)−−−−−→ a−(m−2) → · · · → a−1 d−1−−→ a0 (2.5)

which also fulfils condition (2.4).

• We iterate the previous steps from this new complex.

Note that a0 remains unchanged during the process. Summing up, we have


Lemma 2.12. Let

a−md−m−−−→ a−(m−1) d−(m−1)−−−−−→ a−(m−2) → · · · → a−1 d−1−−→ a0

be a complex of objects of B that fulfil the condition (2.4). There exists a rightconvolution d0 : a0 → a in B, which is uniquely determined up to isomorphism.

When one works with the derived category B = Db(A) of an Abelian categoryand the objects a−p of Db(A) are just objects A−p of the Abelian category A, thenany complex

A• ≡ A−m d−m−−−→ A−(m−1) d−(m−1)−−−−−→ A−(m−2) → · · · → A−1 d−1−−→ A0

fulfils the condition (2.4) and the right convolution a of A• is the complex A•itself, together with the obvious morphism A0 → a = A•.

Lemma 2.13. Let

a−m

f−m

d−m // a−(m−1)

f−(m−1)

d−(m−1) // . . . // a−1

f−1

d−1 // a0

f0

b−m

ed−m // b−(m−1)ed−(m−1) // . . . // b−1

ed−1 // b0

be a morphism between complexes of objects of B fulfilling condition (2.4) and letd0 : a0 → a, d0 : b0 → b be right convolutions. If one has

HomB(a−p[r], b−q) = 0 for every p > q and r > 0 , (2.6)

then for any morphism h : b → b′ there exists a morphism f : a → b′ in B suchthat the diagram

a0d0 //

f0

af

????????

b0ed0 // b

h // b′

(2.7)

is commutative. Moreover, if

HomB(a−p[r], b′) = 0 for every p > 0 and r > 0 , (2.8)

then the morphism f is the only satisfying that property.

Proof. The morphism f is constructed inductively. If m = 0, then f = hf0. Ifm ≥ 1, we have a commutative diagram

a−m //

f−m

a−(m−1)g−(m−1)//

f−(m−1)

a−(m−1) // a−m[1]

f−m[1]

b−m // b−(m−1)

eg−(m−1)// b−(m−1) // b−m[1]


where a−(m−1) and b−(m−1) are cones of the corresponding differentials as before.By the axioms of the triangulated categories, there exists a morphism (not uniquelydetermined) f−(m−1) : a−(m−1) → b−(m−1) completing the diagram. If we considerthe morphisms d−(m−1) : a−(m−1) → a−(m−2) and d−(m−1) : b−(m−1) → b−(m−2)

constructed above, we have

d−(m−1)f−(m−1)g−(m−1) = d−(m−1)g−(m−1)f−(m−1) = d−(m−1)f−(m−1)

= f−(m−2)d−(m−1) = f−(m−2)d−(m−1)g−(m−1)

and then d−(m−1)f−(m−1) = f−(m−2)d−(m−1). Thus, we have a morphism of com-plexes

a−(m−1)

f−(m−1)

d−(m−1)// a−(m−2)

f−(m−2)

d−(m−2) // . . . // a−1

f−1

d−1 // a0

f0

b−(m−1)

ed−(m−1)// b−(m−2)ed−(m−2) // . . . // b−1

ed−1 // b0 .

One easily checks that this morphism of complexes fulfils condition (2.6), andthen we obtain the morphism f : a → b′ by induction. If the condition (2.8) issatisfied, f is uniquely determined by the commutativity of diagram (2.7) becausef0 : a0 → b0 does not change during the process.

Since right convolutions are constructed out of exact triangles and composi-tions of morphisms, they are compatible with exact functors.

Remark 2.14. Let

a−md−m−−−→ a−(m−1) d−(m−1)−−−−−→ a−(m−2) → · · · → a−1 d−1−−→ a0

be a complex of objects of B fulfilling condition (2.4) and let d0 : a0 → a bethe right convolution. If F : B → C is an exact functor to another triangulatedcategory and the complex

F (a−m)F (d−m)−−−−−→ F (a−(m−1))→ · · · → F (a−1)

F (d−1)−−−−−→ F (a0)

of objects of C also fulfils condition (2.4), then its right convolution is F (d0) : F (a0)→ F (a). This happens for instance if F is fully faithful, because in this case

HomΣ(F (a−p)[r], F (a−q)) ' HomB(a−p[r], a−q) = 0 .

4


2.2 Orlov’s representability theorem

This section is devoted to the proof of the following fundamental result by Orlov[242, Thm. 2.2] and some related issues.

Theorem 2.15. Let X and Y be smooth projective varieties. Any fully faithful exactfunctor Ψ: Db(X)→ Db(Y ) is an integral functor.

This is indeed a deep result since in general functors between triangulated cat-egories are quite difficult to describe. It should be pointed out that much strongerresults than Orlov’s theorem hold true in the more flexible setting of dg-categories,which provide an enhancement of the usual derived categories. In this framework,roughly speaking, all functors are integral functors, as Theorem A.57 and, morein particular, Equation A.10 show (see Section A.4.4 and the “Notes and furtherreading” at the end of the current chapter).Remark 2.16. In our proof of Theorem 2.15 we shall use the fact that any exactfunctorDb(X)→ Db(Y ) has a right adjoint (and hence, since the categoriesDb(X)and Db(Y ) have Serre functors, a left adjoint as well). This has been proved byBondal and Van den Bergh [52]. The original result by Orlov was weaker in thatthis property was assumed in the hypotheses of the theorem. 4

2.2.1 Resolution of the diagonal

One of the first important results about derived categories is Beılinson’s construc-tion of a resolution of the structure sheaf of the diagonal of the product Pn × Pnof two copies of the projective space [33, 34], and its implications for the compu-tation of the derived category Db(Pn). We present here Beılinson’s resolution asa particular case of the Koszul sequence associated to certain sections of a locallyfree sheaf.

Let E be a locally free sheaf of rank n on an algebraic variety X and e : OX →E a global section. The zero locus of e is the closed subvariety Z of X defined bythe exact sequence

E∗ e∗−→ OX → OZ → 0 .

We have a Koszul complex

0→n∧E∗ ie−→

n−1∧E∗ ie−→ . . .

ie−→ E∗ e∗−→ OX → OZ → 0 (2.9)

where ie is the inner product with e, i.e., for every affine open subset U andsections e1, . . . , ep on U of E and Ωp on U of

∧p E∗, we have

(ieΩp)(e1, . . . , ep) = Ωp(e, e1, . . . , ep) .

As a consequence of the theory of Koszul complexes we get the following result.

2.2. Orlov’s representability theorem 45

Corollary 2.17. Assume that every point x of Z has an affine neighborhood inX where E∗ has a local basis (ω1, . . . , ωn) such that (ω1(e), . . . , ωn(e)) is a regularsequence in OX(U). Then the Koszul complex (2.9) is exact, thus providing a finiteresolution of the structure sheaf OZ of the zero locus of e by locally free sheaves.

Now take X = Pn × Pn and write Pn as the projective spectrum of thesymmetric algebra Sym(V ), where V is a k-vector space of dimension n + 1. Letus write for simplicity O(m) = OPn(m) and Ω = ΩPn . One has the Euler exactsequence

0→ Ω(1) α−→ V ⊗k O → O(1)→ 0 (2.10)

and taking duals

0→ O(−1)→ V ∗ ⊗k Oα∗−−→ Ω∗(−1)→ 0 .

This gives Γ(Pn,O(1)) ' V and Γ(Pn,Ω∗(−1)) ' V ∗ so that

Γ(X,π∗1O(1)⊗ π∗2Ω∗(−1)) ' V ⊗k V∗ ' End(V ) ,

where π1, π2 are the projections onto the two factors. It follows that the identityon V defines a global section e of E = π∗1O(1)⊗ π∗2Ω∗(−1) to which we can applythe precedent discussion on Koszul complexes.

If we take a basis (x0, . . . , xn) of V , in the open subset Ui where the ho-mogenous coordinate xj do not vanish, we have affine coordinates y(i)

h = xh/xi.On Ui the morphism α is given by dy

(i)h ⊗ x∗i 7→ xh − y

(i)h xi (h 6= i) and on

Ui × Uj a local basis for E∗ is π∗1(x∗i ) ⊗ π∗2(dy(j)h ⊗ xj). Since the identity e on

V as a section of E is e =∑` π∗1(x`) ⊗ π∗2((δ`h − y(j)

h ∂y

(j)h

⊗ x∗j ), computing themorphism e∗ : E∗ → OX on Ui × Uj we see that the ideal of the zero set Z ofe is generated by π∗1(y(i)

h ) ⊗ 1 − π∗1(y(i)j ) ⊗ π∗2(y(j)

h ) (h 6= j), and then Z is thediagonal ∆ of X = Pn × Pn. Moreover, on Ui ×Ui we get that those elements areπ∗1(y(i)

h )⊗1−1⊗π∗2(y(i)h ), and they form a regular sequence. Corollary 2.17 implies

the existence of a resolution of the diagonal, usually called Beılinson resolution ofthe diagonal of the projective space.

Proposition 2.18. There is an exact sequence

0→ π∗1O(−n)⊗ π∗2Ωn(n)→ π∗1O(−(n− 1))⊗ π∗2Ωn−1(n− 1)→ . . .

→ π∗1O(−1)⊗ π∗2Ω(1)→ OXδ∗−→ O∆ → 0 .

This provides a resolution of the structure sheaf of the diagonal ∆ → X = Pn×Pnby locally free sheaves.


Let j > 0 be a positive integer and consider the exact sequence

0→ π∗1O(−n)⊗ π∗2Ωn(n+ j)→ π∗1O(−(n− 1))⊗ π∗2Ωn−1(n− 1 + j)→ . . .

→ π∗1O(−1)⊗ π∗2Ω(1 + j)→ π∗1O ⊗ π∗2O(j) δ∗−→ O∆ ⊗ π∗2O(j)→ 0 ,

obtained as the tensor product of the Beılinson resolution by π∗2O(j). Since all thesheaves Ωp(p + j) (0 ≤ p ≤ n) are acyclic, after taking direct images by π1 wehave an exact sequence

0→ O(−n)⊗k H0(Pn,Ωn(n+ j))→O(−(n− 1))⊗k H

0(Pn,Ωn−1(n− 1 + j))→· · · → O(−1)⊗k H

0(Pn,Ω(1 + j))

→ O⊗k H0(Pn,O(j))→ O(j)→ 0 ,

so that there is an exact sequence

0→ O(−j)→ V j0 ⊗k O → V j1 ⊗k O(1)→ . . .

→ V jn−1 ⊗k O(n− 1)→ V jn ⊗k O(n)→ 0 (2.11)

where V jp = H0(Pn,Ωp(p+ j))∗. We shall use this resolution of O(−j) later on.

Beılinson’s resolution has been generalized by Kapustin, Kuznetsov and Orlovin [171] and further formalized and studied by Kawamata [176, Theorem 3.2]. Weshall consider here Kawamata’s formulation in a way which is sufficient to our pur-poses. In order to state and prove this generalization we need some preliminaries.Let A = ⊕n∈NAn be a graded k-algebra, with A0 = k. Let us define recursivelyvector spaces Bn, with n ∈ N, as the kernels

Bn = ker(Bn−1 ⊗A1 → Bm−2 ⊗A2)

for n ≥ 2, and Bn = An for n = 0, 1. There are natural homomorphisms

Bn ⊗A[−n]→ Bn−1 ⊗A[−n+ 1]

where A[n] is the shifted module A[n]m = An+m.

Definition 2.19. The graded algebra A is Koszul if the sequence

· · · → Bn ⊗A[−n]→ Bn−1 ⊗A[−n+ 1]→· · · → B1 ⊗A[−1]→ A→ k→ 0 (2.12)

is exact. 4


A line bundle L on a projective variety X is said to be Koszul if its associatedhomogeneous coordinate ring

A =⊕n∈N

H0(X,nL)

is Koszul. Theorem 2 in [33] implies that Lk is Koszul for k big enough if L isample.

By composing the morphisms Bn → Bn−1 ⊗A1 (tensored by OX) and A1 ⊗OX → L, one has morphisms ψn : Bn⊗OX → Bn−1⊗L. So we can define sheavesRn on X (with n ∈ N) as Rn = kerψn. We have an exact sequence

0→ Rn → Bn ⊗OX → Bn−1 ⊗ L → · · · → B1 ⊗ Ln−1 → Ln → 0 . (2.13)

Lemma 2.20. [176] There is an exact sequence

0→ A0 ⊗Rn → A1 ⊗Rn−1 → · · · → An−1 ⊗R1 → An ⊗R0 → Ln → 0 . (2.14)

Proof. For any given n ≥ 0 we introduce a double complex of sheaves

Gp,q(n) =

Ap ⊗Bn−p−q ⊗ Lq for p, q, n− p− q ≥ 0

0 otherwise.

The differentials δ1, δ2 are induced by the morphisms in the sequences (2.12) and(2.13). If one studies the associated spectral sequences ′E(n), ′′E(n) one sees thatthe second spectral sequence degenerates at the first step, and

′′Ep,q(n)1 =

Ln for p = 0, q = n

0 otherwise.

Therefore the cohomology of the total complex T • is Hn(T •) = Ln in degree nand 0 otherwise.

The first spectral sequence at first step is

′Ep,q(n)1 =

Ap ⊗Rn−p for q = 0

0 otherwise

and degenerates at the second step. This implies that

ker(Ap ⊗Rn−pδ1−→ Ap+1 ⊗Rn−p−1) = im(Ap−1 ⊗Rn−p+1

δ1−→ Ap ⊗Rn−p)

for p < n, and

Ln ' (An ⊗R0)/ im(An−1 ⊗R1δ1−→ An ⊗R0)

so that the exactness of (2.14) follows.


Proposition 2.21. [176] Let X be a projective variety, ∆ → X ×X the diagonal ofthe product, and let L be a Koszul line bundle on X. Define sheaves Rm for everyinteger m ≥ 0 as before. Then there is an exact sequence

· · · → π∗1L−m ⊗ π∗2Rmdm−−→ π∗1L−(m−1) ⊗ π∗2Rm−1

dm−1−−−→ . . .

d2−→ π∗1L−1 ⊗ π∗2R1d1−→ π∗1OX ⊗ π∗2OX

δ−→ O∆ → 0

of coherent sheaves on X ×X. Moreover, we can choose L so that Hi(X,Ls) = 0for ever i > 0 and s ≥ 1.

Proof. Let us define the complex F•

Fm = π∗1Lm+1 ⊗ π∗2R−m−1 for m ≤ −1

F0 = O∆

Fm = 0 for m > 0 .

Let m0 be a (nonpositive) integer such that Hm0(F•) 6= 0 and let k be an integerwhich is big enough to ensure that

Rpπ2∗(Hq(F•)⊗ π∗1Lk) = 0 for p > 0, q ≥ m0 − dimX

Hp(X,Lk+q) = 0 for p > 0, q ≥ m0 − dimX

R0π2∗(Hm0(F•)⊗ π∗1Lk) 6= 0.

We may associate two spectral sequences to these data. The first has second term

′Ep,q2 = Rpπ2∗(Hq(F•)⊗ p∗1Lk)

and converges to R•π2∗(F• ⊗ π∗1Lk). One has ′Ep,q2 = 0 for p > 0 and q ≥ m0 −dimX while ′E0,m0

2 6= 0, so that Rm0π2∗(F• ⊗ π∗1Lk) 6= 0.

The second sequence has first term

′′Ep,q1 = Rqπ2∗(Fp ⊗ π∗1Lk)

and converges to R•π2∗(F• ⊗ π∗1Lk). One has

′′Ep,q1 =

Lk for p = 0 and q = 0Hq(X,Lk+p+1)⊗R−p−1 for p ≤ −10 otherwise.

As a consequence, ′′Ep,01 = Ak+p+1 ⊗R−p−1 for p ≤ −1 and ′′Ep,q1 = 0 for p+ 1 ≥m0−dimX and q > 0. However Lemma 2.20 implies that ′′Ep,q2 = 0 for p+q = m0,a contradiction.


By reasons that will be evident later on, we need to consider the truncatedcomplex

C•(m) ≡ π∗1L−m ⊗ π∗2Rmdm−−→ . . .

d2−→ π∗1L−1 ⊗ π∗2R1d1−→ π∗1OX ⊗ π∗2OX . (2.15)

Let Tm = ker dm (m ≥ 1) and let αm : Tm[m] → C•(m) be the morphism inducedby the immersion Tm → π∗1L−m ⊗ π∗2Rm. Then the cone Cone(αm) is isomorphicto the complex

Tm → π∗1L−m ⊗ π∗2Rmdm−−→ . . .

d2−→ π∗1L−1 ⊗ π∗2R1d1−→ π∗1OX ⊗ π∗2OX

with Tm at the −(m+ 1)-th place, and is then quasi-isomorphic to O∆.

Assume now that X is smooth and choose m ≥ 2 dimX. Then in the exacttriangle

Tm[m] αm−−→ C•(m) → Cone(αm) ' O∆ → Tm[m+ 1]

the last morphism vanishes, since HomD(X)(O∆, Tm[m+1]) ' Extm+1(O∆, Tm) =0 becausem+1 > 2 dimX andX is smooth. As a consequence, C•(m) is a biproductof O∆ and Tm[m] in the derived category,

C•(m) ' O∆ ⊕ Tm[m] .

Moreover if we call d0 : OX×X → c(m) the convolution of C•(m) in Db(X × X)(which is C•(m) itself, see Section 2.1.2), we can write the above formula in thefollowing form, which we shall shortly use.

c(m) ' O∆ ⊕ Tm[m] . (2.16)

Let us consider now the complex

π∗1Lsk ⊗ C•(m) ' π∗1Lsk−m ⊗ π∗2Rmd′m−−→ . . .

d′2−→ π∗1Lsk−1 ⊗ π∗2R1d′1−→ π∗1Lsk ⊗ π∗2OX , (2.17)

where d′p = 1⊗ dp. By Remark 2.14 we have

π∗1Lsk ⊗ π∗2OXd′0−→π∗1Lsk ⊗ c(m)

'(π∗1Lsk ⊗O∆)⊕ ((π∗1Lsk ⊗ Tm)[m]) ,

where d′0 = 1⊗ d0, is a convolution of π∗1Lsk ⊗ C•(m).


Now fix a value of m (to be specified later) and let s > m. Then all sheavesLsk−p (0 ≤ p ≤ m) are acyclic, so that after applying Rπ2∗ to the complexπ∗1Lsk ⊗ C•(m) we obtain the complex

Rπ2∗(π∗1Lsk ⊗ C•(m)) ' Γ(X,Lsk−m)⊗k Rmπ2∗(d

′m)−−−−−→ . . .

π2∗(d′2)−−−−−→ Γ(X,Lsk−1)⊗k R1

π2∗(d′1)−−−−−→ Γ(X,Lsk)⊗k OX . (2.18)

Again by Remark 2.14,

Γ(X,Lsk)⊗k OXRπ2∗(d

′0)−−−−−−→ Rπ2∗(π∗1Lsk ⊗ c(m))

' Lsk ⊕Rπ2∗(π∗1Lsk ⊗ Tm[m])

is a convolution of Rπ2∗(π∗1Lsk ⊗ C•(m)). Recall that the isomorphism in the for-mula holds when X is smooth and m > 2 dimX − 1.

We finish this section with a lemma that generalizes the argument used inthe proof of (2.16).

Lemma 2.22. Let X be a smooth variety.

1. If E•, F• are objects of Db(X) such that Hi(E•) = 0 for i > z and Hq(F•) =0 for q ≤ z + 2 dimX for an integer z, then HomD(X)(F•, E•) = 0.

2. Let E• be an object of Db(X). If there exist integers z and s > 2 dimX suchthat Hi(E•) = 0 for z < i ≤ z + s, then for every p ∈ [z, z + s] one has anisomorphism

E• ' E•≤p ⊕ E•≥p

in Db(X).

Proof. 1. We proceed by induction on the sum n = `(E•) + `(F•) of the lengthsof E• and F• (recall that the length is the number of nonzero cohomology sheavesof E• and F•). The first case is n = 2, since we can assume that E• and F• arenonzero. Then E• ' E [−q] for q ≤ z and F• ' F [−m] for m > z+ 2 dimX, whereE and F are sheaves. It follows that HomD(X)(F•, E•) ' HomD(X)(F , E [m−q]) 'Extm−qX (F , E) and this is zero because m− q > 2 dimX and X is smooth.

Take n > 2. Then either E• or F• have at least two nonzero cohomologysheaves. If `(F•) ≥ 2, let q0 be the first of the indexes q such that Hq(F•) 6= 0.Then there is a exact triangle

F•≤q0α−→ F• → Cone(α)→ F•≤q0 [1] .


NowHq(F•≤q0) = 0 for q 6= q0 andHq(Cone(α)) = 0 for q ≤ q0, so that they are inthe same hypotheses as F•. Moreover `(E•)+`(F•≤q0) = `(E•)+1 < n and `(E•)+`(Cone(α) = n−1. Then HomD(X)(Cone(α), E•) = 0 and HomD(X)(F•≤q0 , E•) = 0by induction so that HomD(X)(F•, E•) = 0. If `(E•) ≥ 2, we take i0 as the first ofthe integers i such that Hi(E•) 6= 0 and proceed as above using the exact triangle

E•≤i0α−→ E• → Cone(α)→ E•≤i0 [1] .

2. There is a exact triangle

E•≤pα−→ E• → Cone(α) ' E•≥p → E•≤p[1] ,

where the isomorphism Cone(α) ' E•≥p is a consequence of Hp(E•) = 0. MoreoverHi(E•≤p[1]) ' Hi+1(E•≤p) = 0 for i > z and Hq(E•≥p) = 0 for q ≤ z + s ≤z + 2 dimX. Thus HomD(X)(E•≥p, E•≤p[1]) = 0 by the first part, and then E•decomposes as a biproduct in the derived category as claimed.

2.2.2 Uniqueness of the kernel

As a first step in the proof of Orlov’s theorem, and for later use, we want to provethat an integral functor completely determines its kernel. More precisely, let K•be a kernel in D−(X × Y ); we denote simply by Φ the associated integral functorΦK•

X→Y . By using Proposition 1.10 and the base change compatibility of the integralfunctors (Proposition 1.8), we shall prove that K• is completely determined by Φ.

To do so we slightly change notation from Section 1.2. Let us write πij forthe projection of X×X×Y onto its (i, j)-th factor. Then by Proposition 1.10 thekernel K• can be recovered as

K• ' Φπ∗23K•

X×X→X×Y (O∆) = ΦK•X (O∆)

where we have written ΦK•X = Φπ

∗23K•

X×X→X×Y for simplicity.

In the rest of this section we assume that X is smooth and that K• is offinite Tor-dimension as a complex of OX -modules (cf. Section 1.2), so that Φmaps Db(X) into Db(Y ). Then ΦK

•X maps Db(X ×X) into Db(X × Y ).

Lemma 2.23. There exist integer numbers p, m0 such that for every m > m0, onehas a natural isomorphism

K• ' (ΦK•X (c(m)))≥p .

Proof. By (2.16), we have ΦK•X (c(m)) ' ΦK

•X (O∆)⊕ΦK

•X (TM [m]) form ≥ 2 dimX.

Now, Proposition 1.4 for ΦK•X implies the existence of integer numbers z and


n ≥ 0 depending only on Φ such that the cohomology sheaves Hi(ΦK•X (O∆))vanish for i /∈ [z, z + n] and the cohomology sheaves Hi(ΦK•X (Tm[m])) vanish fori /∈ [z−m, z+n−m]. Now take p < z and m0 = max2 dimX, z+n−p. Then thenatural morphism ΦK

•X (O∆) → (ΦK

•X (O∆))≥p is an isomorphism in the derived

category and ΦK•X (c(m)) → ΦK

•X (O∆) induces an isomorphism ΦK

•X (c(m))≥p '

(ΦK•X (O∆))≥p for m ≥ m0 as desired.

Let us prove that ΦK•X (c(m)) depends only on Φ. Since c(m) is a convolution

of C•(m), we consider the complex

ΦK•X (π∗1L−m ⊗ π∗2Rm)

ΦK•X (dm)−−−−−−→ · · · →

ΦK•X (π∗1L−1 ⊗ π∗2R1)

ΦK•X (d1)−−−−−−→ ΦK

•X (π∗1OX ⊗ π∗2OX)

of objects of Db(X × Y ), obtained by applying ΦK•X to the complex C•(m). By

base change (Proposition 1.8) one has isomorphisms

ΦK•X (π∗1L−i ⊗ π∗2Ri) ' π∗1L−i ⊗ π∗2Φ(Ri)

so that one also has a complex

π∗1L−m ⊗ π∗2Φ(Rm)Φ(dm)−−−−→ · · · →

π∗1L−1 ⊗ π∗2Φ(R1)Φ(d1)−−−→ π∗1OX ⊗ π∗2Φ(OX) . (2.19)

The objects of the complex depend only on Φ and not on ΦK•X ; we are going to

show that under appropriate conditions, even the morphisms depend only on Φ.

For every pair of indices p > q and integer r ≥ 0, we have

HomDb(X×Y )(π∗1L−p ⊗ π∗2Φ(Rp)[r], π∗1L−q ⊗ π∗2Φ(Rq))' HomDb(X×Y )(π

∗2Φ(Rp)[r], π∗1Lp−q ⊗ π∗2Φ(Rq))

' HomDb(Y )(Φ(Rp)[r], π2∗(π∗1Lp−q)⊗ Φ(Rq))' HomDb(Y )(Φ(Rp)[r],Γ(X,Lp−q)⊗k Φ(Rq))' HomDb(Y )(Φ(Rp)[r],Φ(Γ(X,Lp−q)⊗k Rq))

(2.20)

where we have used adjunction between inverse and direct images and the projec-tion formula. Analogously one has

HomDb(X×Y )(π∗1L−p ⊗ π∗2Rp[r], π∗1L−(q) ⊗ π∗2Rq)' HomDb(X×Y )(π

∗2Rp[r], π∗1Lp−q ⊗ π∗2Rq)

' HomDb(Y )(Rp[r], π2∗(π∗1Lp−q)⊗Rq)' HomDb(Y )(Rp[r],Γ(X,Lp−q)⊗k Rq) .

(2.21)


Then the natural morphism

HomDb(X)(Rp[r],Γ(X,Lp−q)⊗k Rq)Φ−→

HomDb(Y )(Φ(Rp)[r],Φ(Γ(X,Lp−q)⊗k Rq)) (2.22)

induces a morphism

HomDb(X×Y )(π∗1L−p ⊗ π∗2Rp[r], π∗1L−q ⊗ π∗2Rq)

Φ−→HomDb(X×Y )(π

∗1L−p ⊗ π∗2Φ(Rp)[r], π∗1L−q ⊗ π∗2Φ(Rq)) . (2.23)

Taking in particular q = p− 1 and r = 0, we see that ΦK•X (dp) = Φ(dp). We have

thus proved the following result.

Lemma 2.24. There is an isomorphism

π∗1L−m ⊗ π∗2Φ(Rm)Φ(dm) //

'

. . . // π∗1L−1 ⊗ π∗2Φ(R1)Φ(d1) //

'

π∗1OX ⊗ π∗2Φ(OX)

'

ΦK•X (π∗1L−m ⊗ π∗2Rm)

ΦK•X (dm)// . . . // ΦK

•X (π∗1L−1 ⊗ π∗2R1)

ΦK•X (d1)// ΦK

•X (π∗1OX ⊗ π∗2OX)

of complexes of objects of Db(X ×Y ). As a consequence, the image under ΦK•X of

the complex C•(m) given by (2.15) depends only on Φ and not on ΦK•X .

Assume that Φ is fully faithful. Then (2.23) implies that the complex

π∗1L−m ⊗ π∗2Φ(Rm)Φ(dm)−−−−→ . . .

π∗1−→ L−1 ⊗ π∗2Φ(R1)Φ(d1)−−−→ π∗1OX ⊗ π∗2Φ(OX)

has a convolution, which we denote π∗1OX⊗π∗2Φ(OX) d0−→ (1⊗Φ)(c(m)). Moreover,by Remark 2.14 a convolution of the complex (2.19) exists and is given by

ΦK•X (π∗1OX ⊗ π∗2OX)

ΦK•X (d0)−−−−−−→ ΦK

•X (c(m)) .

Now by Lemma 2.12 there is commutative diagram

π∗1OX ⊗ π∗2Φ(OX)d0 //

'

(1⊗ Φ)(c(m))

'γ

α≥p // ((1⊗ Φ)(c(m)))≥p

'γ≥p


ΦK•X (d0)// ΦK

•X (c(m))

α≥p // (ΦK•X (c(m)))≥p ' K•

for a certain isomorphism γ (not uniquely determined). Here the last isomorphismin the bottom row is due to Lemma 2.23 and the morphisms α≥p are the natural


epimorphisms to the truncations. Since

HomDb(X×Y )(π∗1L−p ⊗ π∗2Φ(Rp)[r],K•)' HomDb(X×Y )(π

∗2Φ(Rp[r]), π∗1Lp ⊗K•)

' HomDb(Y )(Φ(Rp)[r],Rπ2,∗(π∗1Lp ⊗K•))= HomDb(Y )(Φ(Rp)[r],Φ(Lp)) ' HomDb(X)(Rp[r],Lp) = 0

we deduce from Lemma 2.12 that the composed diagonal morphism α≥p γ≥p =α≥p γ : (1⊗Φ)(c(m))→ K• is the unique morphism making the diagram commu-tative. Then, the isomorphism γ≥p is uniquely characterized by the commutativityof the diagram

π∗1OX ⊗ π∗2Φ(OX)α≥pd0 //

'

((1⊗ Φ)(c(m)))≥p

'γ≥p


α≥pΦK•X (d0) // (ΦK

•X (c(m)))≥p ' K• .

(2.24)

This eventually implies the desired uniqueness result. Let fK•

= γ≥ pα≥p d0.

Theorem 2.25. Let X, Y be projective varieties, K• be a kernel in D−(X×Y ) andlet Φ = ΦK

•

X→Y be the corresponding integral functor. Assume that X is smooth,K• is of finite Tor-dimension as a complex of OX-modules and that Φ is fullyfaithful. Then the kernel K• is uniquely determined by the functor Φ. Moreoverif K• is another kernel in D−(X × Y ), of finite Tor-dimension as a complex ofOX-modules, such that Φ = ΦK

•

X→Y , there is a unique isomorphism η : K• ' K• inDb(X × Y ) making the diagram

π∗1OX ⊗ π∗2Φ(OX)fK•

//

fK•

''NNNNNNNNNNNN K•

'η

K•

commutative.

Proof. Since ((1 ⊗ Φ)(c(m)))≥p ' K• one has that K• is uniquely determined byΦ. The second part follows straightforwardly from the above discussion.

2.2.3 Existence of the kernel

In this section we conclude the proof of Orlov’s Theorem 2.15 by constructing thekernel that realizes the given fully faithful functor as an integral functor. Let X,


Y be smooth projective varieties and F : Db(X) → Db(Y ) an exact fully faithfulfunctor. As we shall see in Proposition 2.31, the functor F is bounded, so that thereexist integer numbers z and n ≥ 0 such that for every coherent sheaf F on X, thecohomology sheaves Hi(F (F)) vanish for i /∈ [z, z + n]. Then F can be extendedto a functor D(X)→ D(Y ). We can consider the complex

F (Rπ2∗(π∗1Lsk ⊗ C•(m))) ≡ Γ(X,Lsk−m)⊗k F (Rm)F (π2∗(d

′m))−−−−−−−→ . . .

F (π2∗(d′2))−−−−−−−→ Γ(X,Lsk−1)⊗k F (R1)

F (π2∗(d′1))−−−−−−−→ Γ(X,Lsk)⊗k F (OX) (2.25)

of objects in Db(Y ). By Remark 2.14, the morphism

Γ(X,Lsk)⊗k F (OX)F (π2∗(d

′0))−−−−−−−→ F (Rπ2∗(Lsk ⊗ c(m)))

is a convolution of (2.25). Moreover for m ≥ 2 dimX, (2.16) implies that

F (Rπ2∗(Lsk ⊗ c(m))) ' F (Lsk)⊕ F (Rπ2∗(π∗1Lsk ⊗ Tm))[m] . (2.26)

Let us fix p < z and choose m > maxz + n− p, 2 dimX.

Lemma 2.26. Take s > m such that Rπi2,∗(π∗1Lsk ⊗ Tm) = 0 for i > 0 and k ≥ 1.

Then Hi(F (Rπ2∗(Lsk ⊗ c(m)))) = 0 unless i ∈ [z−m, z+n−m] or i ∈ [z, z+n].Moreover F (Rπ2∗(Lsk ⊗ c(m)))→ F (Lsk) induces an isomorphism

βFk : F (Rπ2∗(Lsk ⊗ c(m)))≥p ' F (Lsk)

in Db(Y ).

Proof. One has

Hi(F (Rπ2∗(Lsk ⊗ c(m)))) ' Hi(F (π2∗(π∗1Lsk ⊗ Tm)))

for i ∈ [z −m, z + n−m] ,

Hi(F (Rπ2∗(Lsk ⊗ c(m)))) ' Hi(F (Lsk)) for i ∈ [z, z + n] ,

Hi(F (Rπ2∗(Lsk ⊗ c(m)))) = 0

for the remaining values of i.

The result follows.

Let us take m > n + 2(1 + dimX + dimY ) so that we can apply Lemma2.26. We consider the truncated complex C•(m) given by (2.15). Since F is fullyfaithful, we have an isomorphism

HomDb(X)(Rp[r],Γ(X,Lp−q)⊗k Rq)F−→

HomDb(Y )(F (Rp)[r], F (Γ(X,Lp−q)⊗k Rq)) .


Then (2.20) remains true if we replace Φ by F , and one obtains, as in Section2.2.2, an isomorphism

F : HomDb(X×Y )(π∗1L−p ⊗ π∗2Rp[r], π∗1L−q ⊗ π∗2Rq) '

HomDb(X×Y )(π∗1L−p ⊗ π∗2F (Rp)[r], π∗1L−q ⊗ π∗2F (Rq)) . (2.27)

Taking in particular q = p − 1 and r = 0, we see that dp induces a morphismF (dp) : π∗1L−p ⊗ π∗2F (Rp)→ π∗1L−(p−1) ⊗ π∗2F (Rp−1) and we have a complex

(1⊗ F )(C•(m)) ≡ π∗1L−m ⊗ π∗2F (Rm)F (dm)−−−−→ · · · →

π∗1L−1 ⊗ π∗2F (R1)F (d1)−−−−→ π∗1OX ⊗ π∗2F (OX) (2.28)

of objects of Db(X×Y ). Furthermore, (2.27) implies that there exists a convolution

π∗1OX ⊗ π∗2F (OX)F (d0)−−−−→ Fc(m) of (1⊗ F ))(C•(m)).

Analogously for any s and k, one has a complex

π∗1Lsk ⊗ (1⊗ F )(C•(m)) ≡ π∗1Lsk−m ⊗ π∗2F (Rm)F (dm)−−−−→ · · · →

π∗1Lsk−1 ⊗ π∗2F (R1)F (d1)−−−−→ π∗1Lsk ⊗ π∗2F (OX) (2.29)

of objects of Db(X × Y ), and π∗1Lsk ⊗ π∗2F (OX)1⊗F (d0)−−−−−→ π∗1Lsk ⊗ Fc(m) is a

convolution of π∗1Lsk ⊗ (1⊗ F )(C•(m)).

Lemma 2.27. 1. There is a (not uniquely determined) isomorphism

ηk : ΦFc(m)X→Y (Lsk) = Rπ2∗(π∗1Lsk ⊗ Fc(m)) ∼→ F (Rπ2∗(π∗1Lsk ⊗ Fc(m)))

in the derived category Db(Y ) which makes the diagram

π2∗(π∗1Lsk ⊗ π∗2F (OX))Rπ2∗(1⊗F (d0)) //

'

Rπ2∗(π∗1Lsk ⊗ Fc(m))

'ηk

= ΦFc(m)X→Y (Lsk)

Γ(X,Lsk)⊗k F (OX)F (π2∗(d

′0)) // F (Rπ2∗(π∗1Lsk ⊗ Fc(m)))

(2.30)commutative.

2. Hi(Fc(m)) = 0 unless i ∈ [z−m, z+n−m] or i ∈ [z, z+n]. Moreover Fc(m)

can be expressed as a biproduct

Fc(m) ' (Fc(m))≥p ⊕ (Fc(m))≤p .


Proof. 1. By applying Rπ2∗ to the complex π∗1Lsk ⊗ (1 ⊗ F )(C•(m)) one obtainsthe complex F (Rπ2∗(Lsk ⊗ C•(m))) described in (2.25). Then, by Remark 2.14,we have an isomorphism ηk : Rπ2∗(π∗1Lsk ⊗ Fc(m)) ∼→ F (Rπ2∗(π∗1Lsk ⊗ Fc(m)))between the convolutions which makes the diagram (2.30) commutative.

2. Assume that there is an integer i not in the prescribed rank such thatHi(Fc(m))) 6= 0. Then for k 0 one has 0 6= π2∗(π∗1Lsk ⊗ Hi(Fc(m))) 'π2∗(Hi(π∗1Lsk⊗Fc(m))) and Rjπ2∗(π∗1Lsk⊗Hi−1(Fc(m))) ' Rjπ2∗(Hi−1(π∗1Lsk⊗Fc(m))) = 0 for every j > 0. There is a spectral sequence Ep,q2 = Rpπ2∗(Hq(π∗1Lsk⊗Fc(m))) converging to Ep+q∞ = Hp+q(Rπ2∗(π∗1Lsk ⊗ Fc(m))). Since E−2,i+1 =E2,i−1

2 = 0, every nonzero element in E0,i2 is a cycle that survives to infinity. Then

E0,i2 6= 0 implies that Hi(Rπ2∗(π∗1Lsk⊗Fc(m))) 6= 0. If we also take the preceding

part 1 into account, this contradicts Lemma 2.26.

Now, since m − n − 2 > 2(dimX + dimY ) and X × Y is smooth, Lemma2.22 gives the decomposition Fc(m) ' (Fc(m))≥p ⊕ (Fc(m))≤p.

Note that if F = ΦK•

X→Y then K• ' (Fc(m))≥p as we saw in Lemma 2.23. Itis therefore convenient in our general situation to define the following objects ofDb(X × Y )

K• ' (Fc(m))≥p , Q• ' (Fc(m))≤p (2.31)

(see Corollary A.35). Then we have an isomorphism of functors

ΦFc(m)X→Y ' ΦK

•

X→Y ⊕ ΦQ•

X→Y .

Lemma 2.28. The natural morphism ΦFc(m)X→Y (Lsk)→ ΦK

•

X→Y (Lsk) induces for everyk an isomorphism

βΦk : (ΦFc(m)

X→Y (Lsk))≥p ' ΦK•

X→Y (Lsk) .

Proof. By Lemma 2.27, one has Hq(Q•) = 0 for q /∈ [z − m, z + n − m]. ThenRpπY ∗(Hq(π∗XLsk ⊗ Q•)) = 0 for q /∈ [z − m, z + n − m + dimX]. Since m >

z+n−p+dimX, one has Hi(ΦQ•X→Y (Lsk)) = 0 for i ≥ p and the result follows.

As a consequence, (2.30) can be completed to a commutative diagram

π2∗(π∗1Lsk ⊗ π∗2F (OX))Rπ2∗(1⊗F (d0)) //

'

ΦFc(m)X→Y (Lsk)

βΦk //

'ηk

fk

))RRRRRRRRRRRRRRRΦK•

X→Y (Lsk)

'γk

Γ(X,Lsk)⊗k F (OX)

F (π2∗(d′0)) // F (Rπ2∗(π∗1Lsk ⊗ Fc(m)))

βFk // F (Lsk)(2.32)


where βFk is the isomorphism given by Lemma 2.26 and

fk = γk βΦk = βFk ηk : ΦFc(m)

X→Y (Lsk)→ F (Lsk) .

Even if ηk is not unique, the morphism γk is, in view of the following lemma.

Lemma 2.29. The isomorphism γk : ΦK•

X→Y (Lsk) ' F (Lsk) in the derived categoryDb(Y ) is uniquely determined by the commutativity of the diagram (2.32). Thus themorphisms γk are functorial, in the sense that, for every morphism α : Lsk → Ls`,there is a commutative diagram in Db(Y )

ΦK•

X→Y (Lsk)ΦK•

X→Y (α) //

'γk

ΦK•

X→Y (Ls`)

'γ`

F (Lsk)

F (α) // F (Ls`) .

Proof. Since

HomDb(Y )(π2∗(π∗1Lsk−p ⊗ π∗2F (Rp))[r], F (Lsk))

' HomDb(Y )(Γ(Y,Lsk−p)⊗k F (Rp))[r], F (Lsk))

' HomDb(X)(Γ(Y,Lsk−p)⊗k Rp[r],Lsk) = 0

as F is fully faithful, Lemma 2.12 implies that fk is the unique morphism whichmakes the diagram (2.32) commutative. Under this condition the morphism γk isunique as well. Functoriality follows straightforwardly.

Since X and Y are smooth, the categories Db(X) and Db(Y ) have Serrefunctors. Moreover Lskk∈Z is an ample sequence in Db(X), and then we canapply Proposition 2.11 to obtain an existence result. The uniqueness of the kernelis given by Theorem 2.25.

Proposition 2.30. Let X, Y be smooth projective varieties and let F : Db(X) →Db(Y ) be an exact fully faithful functor. If F is bounded, then it is an integralfunctor and its kernel is uniquely determined in Db(X × Y ) up to isomorphism.

Orlov’s representability theorem 2.15 follows from Proposition 2.30 due tothe following result:

Proposition 2.31. If X, Y are projective varieties and X is smooth, every exactfunctor F : Db(X)→ Db(Y ) is bounded.


Proof. As we have already noticed, the functor F admits a left adjointG : Db(Y )→Db(X), cf. Remark 2.16. Take an embedding Y → PN induced by a very ample linebundle L. Pulling back to Y the right resolution (2.11), we have a right resolution

0→ L−j → V j0 ⊗k OX → V j1 ⊗k L → · · · →

V jN−1 ⊗k LN−1 → V jN ⊗k LN → 0 (2.33)

for every integer j > 0. Then, for every j > 0 there is a spectral sequence Ep,q2 =Hp(V jq ⊗kG(Lq)) = V jq ⊗kG(Lq)) converging to Ep+q∞ = Hp+q(G(L−j)). Since eachG(Lq) is bounded, there exist integers p0 ≤ p1 such that Hp(G(Lq)) = 0 for everyq ∈ [0, N ] if p /∈ [p0, p1]. This implies that Hk(G(L−j)) = 0 for k /∈ [p0, p1 + N ]and for every j > 0. A similar spectral sequence argument implies that for everysheaf F on X one has

HomiDb(Y )(L

−j , F (F)) = HomiDb(X)(G(L−j),F) = 0 (2.34)

for i /∈ [−p1 − N,−p0 + dimX] and for every j > 0, since X is smooth andthen Homi

Db(X)(G,F) = 0 for any sheaf G on X and any i > dimX. Again, forevery value of j there is a spectral sequence with Ep,q2 = ExtpY (L−j ,Hq(F (F)))converging to Ep+q∞ = Homp+q

Db(Y )(L−j , F (F)). Since F (F) is bounded, for j 0

one has Ep,q2 = 0 for all values of q; then the spectral sequence degenerates yieldingan isomorphism HomY (L−j ,Hq(F (F))) ' Homq

Db(Y )(L−j , F (F)). If Hq(F (F)) 6=

0, we can find j 0 such that HomY (L−j ,Hq(F (F))) 6= 0. Thus, (2.34) impliesthat Hq(F (F)) = 0 unless q ∈ [−p1 −N,−p0 + dimX].

Remark 2.32. Given two kernels K•, G• in Db(X×Y ), any morphism f : K• → G•in the derived category induces a morphism of functors ΦfX→Y : ΦK

•

X→Y → ΦG•

X→Y

which in turn induces another morphism f : K• → G•. This morphism may fail tocoincide with f . Moreover, the groups HomDb(X×Y )(K•,G•) and Hom(ΦK

•

X→Y ,ΦG•X→Y )

may not be isomorphic. In other words, the functor that maps the kernel K• tothe integral functor ΦK

•

X→Y is not fully faithful in general.

Take for instance for X = Y an elliptic curve, K• = O∆ and G• = O∆[2].The Serre functor of X ×X consists in the shift by 2, so that

HomDb(X×X)(O∆,O∆[2]) ' HomDb(X×X)(O∆,O∆)∗ ' k .

On the other hand, ΦK•

X→Y is the identity functor IdDb(X) and ΦG•

X→Y ' IdDb(X)[2].If g : IdDb(X) → IdDb(X)[2] is a functor morphism, then for every sheaf F onX, the induced morphism g(F) : F → F [2] is zero since HomDb(X)(F ,F [2]) =Ext2

X(F ,F) = 0 because X is a curve. One easily proves that the morphism g(F•)is also zero for every bounded complex F•; thus g = 0, that is,

Hom(ΦK•

X→Y ,ΦG•X→Y ) ' Hom(IdDb(X), IdDb(X)[2]) = 0 .


The fact that the functor mapping kernels to integral functors may fail to befully faithful can be regarded as an intrinsic limitation of the triangulated struc-ture of the derived category. Actually, if we pass to the setting of dg-categories,the corresponding functor (suitably defined) turns out to be an equivalence. SeeTheorem A.57 and the comments in “Notes and further reading.”

4

2.3 Fourier-Mukai functors

The following definition introduces the objects that will be our main concern inthis book.

Definition 2.33. An integral functor ΦK•

X→Y : Db(X) → Db(Y ) is called a Fourier-Mukai functor if it is an exact equivalence of derived categories. If in addition thekernel is a concentrated complex, the functor will be said to be a Fourier-Mukaitransform. 4

We give now some basic properties of the Fourier-Mukai functors. Later onwe shall describe some geometric applications of Fourier-Mukai functors and shallestablish a criterion for testing whether an integral functor is a Fourier-Mukaifunctor.

The composition of two Fourier-Mukai functors is a Fourier-Mukai functoras well (and, as we know, the kernel of the composition is the convolution of thetwo kernels, cf. Proposition 1.3). However, the composition of two Fourier-Mukaitransforms may fail to be a Fourier-Mukai transform, because its kernel may notbe a concentrated complex, as we shall see in Example 2.59.

Fourier-Mukai functors behave well with respect to the WIT condition. LetΦ = ΦK

•

X→Y : Db(X) → Db(Y ) be a Fourier-Mukai functor; the functor i-th coho-mology sheaf Φi(•) = Hi(Φ(•)) will be called the i-th Fourier-Mukai functor. Givena quasi-inverse Φ : Db(Y )→ Db(X) of Φ, we have an isomorphism

Φ(Φ(E•)) ' E•

in the derived category. When E• is a sheaf E in degree zero, the above isomorphismmeans that there is a convergent spectral sequence

Ep,q2 = Φp(Φq(E)) =⇒

E if p+ q = 00 otherwise.

(2.35)

Proposition 2.34. If Φ is a Fourier-Mukai functor and E is a WITi sheaf, then the

unique nonzero Fourier-Mukai sheaf E is a WIT−i sheaf. Moreover E = Φ−i(E) 'E.

2.3. Fourier-Mukai functors 61

We shall also need the following result.

Proposition 2.35. Let Φ: Db(X)→ Db(Y ) be a Fourier-Mukai functor and assumethat X is smooth of dimension n. For every (closed) point x ∈ X the followinginequality holds true:∑

i

dim Hom1D(Y )(Φ

i(Ox),Φi(Ox)) ≤ n .

Proof. There is a spectral sequence Ep,q2 =⊕

i HompD(Y )(Φ

i(Ox),Φi+q(Ox)) con-

verging to Ep+q∞ = Homp+qD(Y )(Φ(Ox),Φ(Ox)). The exact sequence of the lower

terms yields 0 → E1,02 → E1

∞. By the Parseval formula (Proposition 1.34), onehas Hom1

D(Y )(Φ(Ox),Φ(Ox)) ' Hom1D(X)(Ox,Ox) ' kn.

2.3.1 Some geometric applications of Fourier-Mukai functors

The existence of a Fourier-Mukai functor between the derived categories of twosmooth algebraic varieties (or the equivalent conditions that the two algebraic va-rieties have equivalent derived categories, cf. Theorem 2.15) imposes strong con-straints on their geometry. A first manifestation of this fact is Theorem 2.38 whichstates that the two (smooth projective) varieties X and Y have the same dimen-sion, and that their canonical bundles satisfy some stringent conditions. Corol-lary 2.40 will establish that the rational Chow rings of X and Y are isomorphic(as Q-vector spaces). According to Theorem 2.49, under a condition on the Ko-daira dimension of X, the varieties X and Y are birational. If the hypotheses arestrengthened by assuming that X has an ample canonical bundle, then X and Y

are isomorphic (Theorem 2.51).

The following definition is commonly adopted for varieties with equivalentbounded derived categories.

Definition 2.36. Two projective varieties X and Y are Fourier-Mukai partners ifthere is an exact equivalence of triangulated categories F : Db(X) ∼→ Db(Y ). 4

Note that we do not impose that any of the varieties is smooth. However, ifone is smooth, the other is smooth as well.

Lemma 2.37. Let X be a smooth projective variety.

1. Every Fourier-Mukai partner of X is smooth.

2. A projective variety Y is a Fourier-Mukai partner of X if and only if thereis Fourier-Mukai functor ΦK

•

X→Y : Db(X) ∼→ Db(Y ).


Proof. Assume that X is a smooth variety and that there is an equivalence ofcategories F : Db(Y ) → Db(X). For every (closed) point y ∈ X and every sheafF on Y , one has Homi(Oy,F) ' Homi(F (Oy), F (F)). Since X is smooth andF (Oy) and F (F) are bounded, there is only a finite number of indexes i withHomi(Oy,F) 6= 0. Then Oy is of finite homological dimension, and hence Y issmooth at y by Serre’s Theorem [266], [215, 19.2]. This proves the first part. Thesecond follows from Orlov’s representability theorem 2.15.

We briefly recall the notion of determinant bundle for an objectM• of Db(X)where X is a smooth projective variety. This is defined by

det(M•) =⊗i

(det(E i))(−1)i

where E• is any bounded complex of locally free sheaves isomorphic toM• in thederived category and det(E i) is the highest exterior power of E i.

A direct computation shows that

det(M• ⊗ L) = det(M•)⊗ Lrk(M•) (2.36)

for every line bundle L.

Theorem 2.38. Let X, Y be smooth projective varieties that are Fourier-Mukaipartners, so that there is a Fourier-Mukai functor ΦK

•

X→Y : Db(X)→ Db(Y ).

1. The right and left adjoints to ΦK•

X→Y are functorially isomorphic

ΦK•∨⊗π∗XωX [m]

Y→X ' ΦK•∨⊗π∗Y ωY [n]

Y→X

(here m = dimX and n = dimY ) and they are both quasi-inverses to ΦK•

X→Y .

2. X and Y have the same dimension, m = n.

3. ωX and ωY have the same order, that is, ωkX is trivial if and only if ωkY istrivial. Thus, ωX is trivial if and only if ωY is trivial and in this case thefunctor ΦK

•∨[n]Y→X is a quasi-inverse to ΦK

•

X→Y .

4. ωrX ' OX and ωrY ' OY with r = rk(K•).

Proof. 1. Since a quasi-inverse is both a right and a left adjoint, the uniqueness ofadjoints together with Proposition 1.13 yields the statement.

2. Applying the above functorial isomorphism to the skyscraper sheaf Oywe obtain Lj∗yK•∨ ⊗ ωX [m] ' Lj∗yK•∨[n]. Since the functors we have applied areequivalences of categories, both objects are nonzero in Db(X). Then there is an


integer q0 which is the minimum of the q’s with Hq(Lj∗yK•∨) 6= 0 and one getsq0 +m = q0 + n so that m = n.

3. Assume for instance that ωkX is trivial; the other case is proved analogously.By Corollary 1.18, one has SkY ' ΦK

•

X→Y SkX (ΦK•

X→Y )−1, where (ΦK•

X→Y )−1 is aquasi-inverse to ΦK

•

X→Y . Since ωkX is trivial, SkX(F•) ' F•[kn] and then SkY 'ΦK•

X→Y [kn] (ΦK•

X→Y )−1 ' [kn]. Therefore ωkY ' OY .

4. Taking determinant bundles in the expression Lj∗yK•∨⊗ωX ' Lj∗yK•∨, wehave det(Lj∗yK•∨) ' det(Lj∗yK•∨)⊗ωryX with ry = rk(Lj∗yK•∨) by Equation (2.36),and therefore ωryX ' OX . Now the functoriality of the Chern classes gives ry =rk(K•∨) = r. The proof of the second formula is analogous.

As we recalled at the beginning of this chapter, when K• is a sheaf Q con-centrated in degree zero the dual complex may be different from the concentratedcomplex given by the dual sheaf Q∗ in degree zero. This point deserves a comment.

Example 2.39. Take K• = O∆, the structure sheaf of the diagonal ∆ ⊂ X × X.The integral functor ΦO∆

X→X is isomorphic to the identity functor as we have seenin Example 1.2. We have that O∗∆ = 0 because O∆ is a torsion sheaf, so thatΦO

∗∆⊗π

∗1ωX [n]

X→X = 0 and this cannot be a quasi-inverse to ΦO∆X→X . The identity functor

is of course a quasi-inverse to itself, and according to Theorem 2.38 it must coincidewith ΦO

∨∆⊗π

∗i ωX [n]

X→X for i = 1, 2. Let us check that this is indeed the case. SinceX is smooth, the diagonal is a regular embedding and then a standard localcomputation using the Koszul complex yields the formulas

ExtiOX×X (O∆,OX×Y ) '

0 for q 6= n

δ∗(ω∗X) for q = n.(2.37)

Thus, O∨∆ ' δ∗(ωX)[−n] in the derived category, and therefore O∨∆ ⊗ π∗i ωX [n] 'O∆ so that ΦO

∨∆⊗π

∗i ωX [n]

X→X ' ΦO∆X→X is the identity.

Looking at things the other way round, one should say that Theorem 2.38together with the uniqueness of the kernel (Theorem 2.25) proves O∨∆⊗π∗i ωX [n] 'O∆ and therefore yields the formula 2.37 without resorting to the Koszul complex.

4

Theorem 2.38 allows us to prove an important property of the map fK•

defined in Equation (1.12).

Corollary 2.40. Let X, Y be smooth projective varieties and let ΦK•

X→Y : Db(X) →Db(Y ) be a Fourier-Mukai functor. The induced map fK

•: A•(X)⊗Q→ A•(Y )⊗Q

is an isomorphism of Q-vector spaces. Moreover, if k = C, the induced f -map incohomology fK

•: H•(X,Q)→ H•(Y,Q) is also an isomorphism of Q-vector spaces


which induces an isomorphism of vector spaces between the even cohomology rings.

Proof. By Theorem 2.38, ΦK•∨⊗π∗XωX [m]

Y→X is a quasi-inverse to ΦK•

X→Y , so that theconvolution K• ∗ (K•∨ ⊗ π∗XωX [m]) is isomorphic to O∆ in the derived categorybecause of the uniqueness of the kernel (Theorem 2.25). The functoriality of themap f (cf. Eq. (1.13)) yields fK

• fK•∨⊗π∗XωX [m] = fO∆ . Since v(O∆) = δ∗(1)by Grothendieck-Riemann-Roch for the diagonal immersion δ, we have fO∆ = Id(here v is the Mukai vector defined in Eq. (1.1)). One analogously proves thatfK•∨⊗π∗XωX [m] fK• = Id. The cohomology statement is proved in a similar way.

Part 4 of Theorem 2.38 implies that whenever the kernel K• is not of rankzero, a certain power ωrX of the canonical bundle of X has to be trivial, withr 6= 0. This is a strong geometric constraint: if X is a curve, it has to be elliptic(and then ωX ' OX); if X is a surface, it has to be Abelian, K3 (in which casesωX ' OX), Enriques (for which ω2

X ' OX) or bielliptic (for which ω12X ' OX)

(cf. [141, Thm. 6.3]). In dimension 3 the most important example is provided byCalabi-Yau varieties (for which, by definition, ωX ' OX). This is the reason whythe Fourier-Mukai transform has been mostly studied for this kind of variety.

However, this by no means implies that if all powers ωrX (with nonzero expo-nent) are nontrivial, then Fourier-Mukai functors ΦK

•

X→Y : Db(X)→ Db(Y ) cannotexist. Rather they do exist, but the kernel K• must be of rank zero, as in the caseof the structure sheaf of the diagonal.

We shall deal with the case of rank zero kernels when in Chapter 6 we shallstudy integral transforms for families, or relative integral transforms. Given twofamilies of varieties X → S and Y → S, we shall define an integral functorΦK•

X→Y : Db(X) → Db(Y ) by means of a relative kernel K• in the derived categoryDb(X ×S Y ) of the fiber product. That transform will be defined as the ordinaryintegral functor with kernel i∗K•, where i : X ×S Y → X × Y is the naturalimmersion. Even when the integral functor ΦK

•

X→Y is an equivalence of categorieswe cannot use Theorem 2.38 to get information about ωX , because as a complexin Db(X×Y ) we have rk(i∗K•) = 0. Here one needs a relative version of Theorem2.38 where the relative canonical sheaves ωX/S and ωY/S replace the absolutecanonical sheaves.

Let X, Y be smooth projective varieties and ΦK•

X→Y a Fourier-Mukai functor.By Theorem 2.38, a quasi-inverse is given by ΦK

•∨⊗π∗XωXY→X = ΦK

•∨⊗π∗Y ωYY→X . Let W

and W∨ be the supports of K• and K•∨, respectively (see Definition A.90).

Proposition 2.41. One has W = W∨ and the two projections πX |W : W → X,πY |W : W → Y are surjective.


Proof. There is a convergent spectral sequence with Ep,q2 = ExtpOX×Y (Hp(K•),O)and Ep+q∞ = Hp+q(K•∨). This proves that W∨ ⊆ W . Reversing the roles of K•and K•∨ we get that W ⊆ W∨. Now, since ΦK

•

Y→X is an equivalence, ΦK•

Y→X(Ox) =Lj∗xK• 6= 0 for every x ∈ X; this proves that the morphism πX |W : W → X issurjective. The surjectivity of πY |W : W → Y is proved analogously by using thefact that ΦK

•∨

Y→X is also a Fourier-Mukai functor by Theorem 2.38.

The existence of a Fourier-Mukai functor ΦK•

X→Y : Db(X) → Db(Y ) (which isequivalent to the existence of an exact equivalence by Theorem 2.15) has inter-esting effects on the geometry of X and Y . Our next aim is to prove some strongresults in that direction due to Orlov and Kawamata

We begin by recalling some standard definitions. If X is a smooth projectivevariety, and L is a line bundle on X, then for n 0 the dimension of the globalsections Γ(X,Ln) of Ln is a polynomial in n of a certain degree d ≤ dimX; thedegree of the null polynomial is −∞ by decree. The degree of such polynomial iscalled the Kodaira dimension of L and it is denoted by κ(X,L). So one knowsthat κ(X,L) ≤ dimX.

In particular, the Kodaira dimension of X is defined as κ(X) = κ(X,ωX).One can also define the Kodaira dimension in terms of the projective rational mapsdefined by Ls (s ≥ 0), assuming that they exist. In that case (which correspondsto κ(X,L) > −∞), κ(X,L) is the maximum of the dimensions of the images ofthose maps, and it is also the transcendence degree of the graded ring R(X,L) =⊕s≥0Γ(X,Ls) minus 1.

Lemma 2.42. (Kodaira’s Lemma) If κ(X) = dimX (resp. κ(X,ω∗X) = dimX),there exist an ample divisor H and an integer s0 such that for any integer s ≥ s0

there is an effective divisor Ds such that ωsX ' OX(H)⊗OX(Ds) (resp. (ω∗X)s 'OX(H)⊗OX(Ds)).

Proof. Assume that κ(X) = dimX. Let H → X be a smooth ample divisor, whichexists by Bertini’s theorem [141, 8.18], and consider the exact sequence

0→ ωsX(−H)→ ωsX → ωXs|H → 0 .

Since χ(X,ωsX) is a polynomial in s of degree κ(X) = dimX, and χ(H,ωXs|H) isa polynomial in s of degree κ(H,ωX |H) ≤ dimH = dimX − 1, we see that fors 0 the line bundle ωsX(−H) has a section. Thus, ωsX(−H) ' OX(Ds) for someeffective divisor Ds. The other case is analogous.

Recall that a line bundle L on a projective variety is numerically effectiveor nef if for any morphism φ : C → X where C is a projective curve, one hasdeg φ∗L ≥ 0. We can consider only closed immersions C → X, because we canalways replace φ : C → X by its image.


Lemma 2.43. Let f : Y → X be a projective morphism.

1. If L is a nef line bundle on X, then f∗L is nef on Y .

2. If f is surjective, then a line bundle L on X is nef if and only if f∗L is nefon Y .

Proof. The first claim is obvious. For the second, let C be a projective curve,φ : C → X a morphism and N a very ample line bundle for the projective mor-phism fC : C ×Y X → C. For n 0, N n has a section which defines a divisorH → C ×Y X. The curve C = Hr (r = dimY − dimX) intersects every fiber ina finite number of points, so that the projection π : C → C is a finite morphism.Moreover the composition φ π : C → X factors as f ρ, where ρ : C → Y is theinduced morphism. Since f∗L is nef, deg ρ∗f∗L ≥ 0, then deg φ∗L ≥ 0 as well.

We can define the numerical Kodaira dimension of a line bundle L on aprojective variety as the maximum ν(X,L) of the integer numbers m such thatthere is a proper morphism ϕ : T → X from a variety T of dimension m withthe property ϕ∗(c1(L))m · T 6= 0. The intersection numbers ϕ∗(c1(L))m · T can bedefined in terms of the Snapper polynomial. To this end, let us recall that for anyline bundle N on a m-dimensional projective variety T , the Euler characteristicχ(T,Nn) of Ln is a polynomial in n of a certain degree d ≤ m, called the Snapperpolynomial [119, Ex. 18.3.6], and that

χ(T,Nn) =1m!c1(N )m · T nm + terms of lower degree.

It is clear that we can define the numerical Kodaira dimension of L by consideringonly closed immersions ϕ : T → X. Moreover, the numerical Kodaira dimensionof any power of a line bundle L equals that of L, namely, ν(X,L) = ν(X,Ls) forany s 6= 0.

When L is nef, the numerical Kodaira dimension is the maximum of theintegers m such that c1(L)m is not numerically trivial. In this case, the numericalKodaira dimension is bounded by the Kodaira dimension, ν(X,L) ≤ κ(X,L).

If X is a projective Gorenstein variety, the numerical Kodaira dimension ofX is defined as ν(X) = ν(X,ωX).

Lemma 2.44. Let f : Y → X be a projective morphism and L a line bundle onX. Then ν(Y, f∗L) ≤ ν(X,L). Moreover, if f is surjective, one has ν(Y, f∗L) =ν(X,L).

Proof. The first claim is obvious. The proof of the second is similar to that ofLemma 2.43. Let ϕ : T → X be a proper morphism such that ϕ∗(Lm) ·T 6= 0 with


m = dimT , and let N be a very relatively ample line bundle for the projectivemorphism fT : T ×Y X → T . For n 0, Nn has a section which defines a divisorH → T ×Y X. Then T = Hr (r = dimY − dimX) intersects every fiber in afinite number of points, so that the projection π : T → T is a finite morphism.Moreover the composition ϕ π : T → X factors as f ρ, where ρ : T → Y is theinduced morphism. It follows that ρ∗(f∗Lm) · T ' π∗(ϕ∗Lm) · T 6= 0, and thenν(X,L) ≤ ν(Y, f∗L), so that ν(Y, f∗L) = ν(X,L) as claimed.

Finally, we need a technical result whose proof we include although it isstandard.

Lemma 2.45. Let Z be a normal variety and F a rank r coherent sheaf on Z. IfL1 and L2 are line bundles on Z such that F ⊗ L1 ' F ⊗ L2, then Lr1 ' Lr2.

Proof. Modding the torsion out we can assume that F is torsion-free. Since Z isnormal there is a codimension two closed subset Z ′ such that F is locally free ofrank r on U = Z − Z ′. By the theorem on generic smoothness [141, III.10.7], Ucan be assumed to be smooth. Taking determinants, we get det(F|U ) ⊗ Lr1|U 'det(F|U )⊗Lr2|U and thus Lr1|U ' Lr2|U . Since Z is normal and Z ′ has codimension2, this isomorphism can be extended to an isomorphism Lr1 ' Lr2 (cf. [140, Theorem3.8]).

Let X and Y be smooth projective varieties and ΦK•

X→Y a Fourier-Mukai func-tor. For every irreducible component Z of the support W of K•, we denote byZ → Z its normalization and by pX : Z → X, pY : Z → Y the induced maps.

Lemma 2.46. One has p∗XωrX ' p∗Y ω

rY for some r > 0. In particular p∗XKX and

p∗YKY are Q-linearly equivalent. Moreover, we can chose an irreducible compo-nent ZX(K•) of W such that pX = πX |ZX(K•) : ZX(K•) → X and then alsopX : ZX(K•)→ X, are dominant.

Proof. By Theorem 2.38, one has dimY = dimX and if we denote by n thisdimension, a quasi-inverse to ΦK

•

X→Y is given by ΦK•∨⊗π∗XωX [n]

Y→X ' ΦK•∨⊗π∗Y ωY [n]

Y→X . Theuniqueness of the kernel (Theorem 2.25) implies that K•∨⊗π∗XωX ' K•∨⊗π∗Y ωY ,so that Hi(K•∨)⊗ π∗XωX ' Hi(K•∨)⊗ π∗Y ωY for every i. If ρ : Z → X × Y is thecomposition of Z → Z and the immersion Z → X × Y , we see that ρ∗(Hi(K•∨)⊗p∗XωX) ' ρ∗(Hi(K•∨) ⊗ p∗Y ωY ). By Lemma 2.45, p∗Xω

rX ' p∗Y ω

rY where r is the

rank of Hi(K•∨)|Z , which is not zero.

By Proposition 2.41, πX |W : W → X is surjective. Then we can choose anirreducible component ZX(K•) of W which dominates X.


The following Proposition 2.48 and Theorem 2.49 express a result due toKawamata, usually known as “D-equivalence implies K-equivalence.” We nowgive the precise definition of K-equivalent algebraic varieties.

Definition 2.47. Two smooth projective algebraic varieties X and Y are K-equiva-lent if there are a normal variety Z and projective birational morphisms pX : Z →X, pY : Z → Y such that p∗XKX and p∗YKY are Q-linearly equivalent, that is,rp∗XKX and rp∗YKY are linearly equivalent for some r 6= 0. 4

Notice that if F• is a simple object of Db(Y ), i.e., HomD(Y )(F•,F•) = k, thesupport of F• is connected. This can be seen as follows: assume that SuppF• =Y1

∐Y2; if we represent F• as a finite complex E• of locally free sheaves of finite

rank, the natural map E• → j1∗j∗1E• ⊕ j2∗j

∗2E• is a quasi-isomorphism. Then

F• ' j1∗j∗1E• ⊕ j2∗j∗2E• in the derived category, so that F• is not simple.

Proposition 2.48. Let ZX(K•) be an irreducible component of W = SuppK• suchthat pX : ZX(K•)→ X is dominant.

1. There is a nonempty open subset U of X such that p−1X (x) = π−1

X (x)∩W 'Supp ΦK

•

X→Y (Ox) for every point x ∈ U .

2. If dimZX(K•) = dimX, then pX : ZX(K•)→ X and pY : ZX(K•)→ Y arebirational and p∗XKX and p∗YKY are Q-linearly equivalent. That is, X andY are K-equivalent.

Proof. If Z1, . . . , Zs are the irreducible components of W other than Z = ZX(K•),and T = ∪i(Z ∩ Zi), we take U = X − pX(T ). Since ΦK

•

X→Y (Ox) is simple, itssupport Supp ΦK

•

X→Y (Ox) ' π−1X (x) ∩W is connected as we have just seen. Then

Supp ΦK•

X→Y (Ox) has to be contained in π−1X (x) ∩ Z = p−1

X (x) if x ∈ U ; this provesthe first part.

We now prove the second part. We have to prove that both pX and pY arebirational. Consider first the projection pX : Z → X. Since dimZ = dimX, pXis generically finite. By Zariski’s main theorem [141, 11.4], to prove that it isbirational, we need only to check that it is generically injective. By the first part,p−1X (x) = π−1

X (x) ∩W for generic x ∈ X, and then π−1X (x) = (x, y1), . . . , (x, ys)

is finite. Thus, Lj∗xK• = ΦK•

X→Y (Ox) is supported at the points y1, . . . , ys, andHomDb(Y )(ΦK

•

X→Y (Ox),ΦK•

X→Y (Ox)) is of dimension s. By the Parseval formula (cf.Proposition 1.34), one has

HomDb(Y )(ΦK•X→Y (Ox),ΦK

•

X→Y (Ox)) ' HomDb(X)(Ox,Ox) = k ,

so that s = 1. This proves that pX is generically injective.

Our last step is to show that pY : Z → Y is also birational. If we prove that itis dominant, taking into account that Z is also an irreducible component of W∨ =


SuppK•∨ (cf. Proposition 2.41), we deduce as above that pY is birational. Since pXis birational, there is an open subset U ′ ⊆ U such that πX induces an isomorphismbetween p−1

X (U ′) = π−1X (U ′) ∩W and U ′. If we assume that pY is not dominant,

then dim pY (Z) < dimY and there exist distinct points x1, x2 of U ′ such thaty = πY (π−1

X (x1)∩W ) = πY (π−1X (x2)∩W ). Thus, the point y is the support of both

ΦK•

X→Y (Ox1) and ΦK•

X→Y (Ox2), so that HomiDb(Y )(Φ

K•X→Y (Ox1),ΦK

•

X→Y (Ox2)) 6= 0 forsome integer i. By the Parseval formula, this implies that Homi

Db(X)(Ox1 ,Ox2) 6=0, which is absurd.

Finally, since p∗XωrX = p∗Y ω

rY , we eventually obtain that p∗XKX and p∗YKY

are Q-linearly equivalent.

Theorem 2.49. [175, 176] Let X, Y be smooth Fourier-Mukai partners.

1. The line bundle ωX (resp. ω∗X) is nef if and only if ωY (resp. ω∗Y ) is nef.

2. X and Y have the same numerical Kodaira dimension, ν(X) = ν(Y ).

3. If the Kodaira dimension κ(X) is equal to dimX (or if κ(X,ω∗X) = dimX),then X and Y are K-equivalent.

Proof. By Lemma 2.37, there is a Fourier-Mukai functor F = ΦK•

X→Y . Let ZX(K•) bean irreducible component of the support W of K• which dominates X (cf. Lemma2.46; we use the same notation as in this lemma).

1. If ωY is nef so is ωrY , then p∗Y ωrY is nef by Lemma 2.43 and since pX is

surjective and p∗XωrX ' p∗Y ω

rY , the same lemma implies that ωrX , and hence ωX ,

is nef. The case when ω∗Y is nef is proved analogously.

2. By Lemma 2.44, ν(X,ωX) = ν(ZX(K•), p∗XωX) = ν(ZX(K•), p∗Y ωY ) ≤ν(Y, ωY ). Reversing the roles of X and Y one proves the converse statement.

3. Assume first that κ(X) = dimX. By Kodaira’s Lemma 2.42, one can takem > 0 such that the two isomorphisms p∗Xω

mX ' p∗Y ω

mY and ωmX ' OX(H) ⊗

OX(D), where H is an ample divisor and D is effective, hold true.

Let us see that pY : ZX(K•) → Y is quasi-finite (i.e., it has finite fibers)outside p−1

X (D). First note that since pX and pY are the restrictions to ZX(K•) ofthe projections of X × Y onto its factors, no curve can be contracted by both ofthem; since normalization is a finite morphism, the same happens for pX and pY .Assume now that there is a curve C contained in a fiber p−1

Y (y) and not entirelycontained in p−1

X (D); then we have p−1Y KY · C = 0 and

mp−1Y KY · C = mp−1

X KX · C = p−1X H · C + p−1

X D · C ≥ p−1X H · C .

Since C cannot be contracted by pX and H is ample, we get mp−1Y KY · C > 0,

which is a contradiction. Thus dimZX(K•) = dim ZX(K•) ≤ n = dimY so that


dimZX(K•) = n, and we conclude by Proposition 2.48. The case κ(X,ω∗X) =dimX is analogous.

Remark 2.50. The variety ZX(K•) in Lemma 2.46 may be assumed to be smoothpossibly by replacing it with a resolution of its singularities. 4

A consequence of Kawamata’s Theorem 2.49 is a celebrated “reconstructiontheorem” due to Bondal and Orlov [49].

Theorem 2.51. Let X, Y be smooth Fourier-Mukai partners. If either ωX or ωYis ample or anti-ample, there is an isomorphism X ' Y .

Proof. Assume that ωX is ample or anti-ample. If C → ZX(K•) is a curve con-tracted by pY , then p∗Xω

mX |C ' p

∗Y ω

mY |C ' OC , which is impossible because ωX is

ample or anti-ample and C is not contracted by pX as we have seen in the proofof Theorem 2.49. Hence pY is an isomorphism and we have a birational morphismf : Y → X of smooth varieties such that f∗ωmX ' ωmY with m > 0 when ωX isample. Let us now prove that f is actually an isomorphism. In the exact sequenceof differentials

f∗ΩXdf−→ ΩY → ΩY/X → 0

the morphism df is injective (because it is injective at the generic point and f∗ΩXis locally free); then ΩY/X is a torsion sheaf supported by a closed subschemeY ′ 6= Y which coincides with the zeroes of the determinant of df . This determinantis a section of ωY ⊗ f∗ω−1

X . Then (det(df))m is a section of (ωY ⊗ f∗ω−1X )m ' OY

vanishing on Y ′. Thus Y ′ is empty, so that ΩY/X = 0 and f is smooth of relativedimension zero, and being also birational, it is an isomorphism.

We have chosen to give a detailed account of this proof of this importanttheorem because the techniques introduced and its underlying ideas will be use-ful elsewhere in this book, in particular when we shall study the Fourier-Mukaipartners of a variety. However, the original proof by Bondal and Orlov in [49] ismore direct and does not make use of Orlov’s representability theorem 2.15. Italso enlightens how the derived category of a variety encodes information aboutits points and the line bundles on it. For this reason we offer here a brief sketchof that proof.

One starts by defining the point objects and invertible objects in a triangu-lated category. One shows that when the latter is the derived category Db(X) ofcoherent sheaves of a smooth algebraic variety with ample canonical or anticanon-ical sheaf, the point objects are the complexes of the form Ox[i] with i ∈ Z. Onthe other hand, when point objects have this form, one shows that the invertibleobjects are the objects of the form L[i] with i ∈ Z, where L is a line bundle.

Assume now that X is a smooth projective algebraic variety with ample oranti-ample canonical bundle, and Y is a Fourier-Mukai partner ofX, i.e., that there


is an exact equivalence of triangulated categories F : Db(X)→ Db(Y ). One provesthat F maps point objects to point objects, and this in turn implies that the pointobjects in Y are again exactly the shifted skyscraper sheaves. Along the same lines,one proves that F maps invertible objects to invertible objects. The result aboutpoint objects implies now that line bundles are mapped into shifted line bundles.By suitably redefining the functor F , one can obtain that skyscraper sheaves aremapped to skyscrapers, thus providing a set-theoretic identification of X with Y ,and that line bundles are mapped to line bundles. The latter property impliesthat X and Y are homeomorphic. After redefining F again, we can assume thatF (OX) ' OY , and since F commutes with the Serre functor, one has F (ωiX) ' ωiYfor all i. Since ωX is ample or anti-ample, the topology of X has a basis formedby open sets Uα labeled by α ∈ HomX(ωiX , ω

jX), i, j ∈ Z; by definition, Uα is the

set of points in X where α does not vanish. The properties of F that one has sofar proved imply that the open sets Vα ⊂ Y defined in the same way in terms ofωY are also a basis of the topology of Y . A theorem by Illusie [158] implies thatωY is ample or anti-ample, respectively.

Moreover, the equivalence F induces an isomorphism between the gradedcanonical algebras ⊕i≥0HomX(OX , ωiX) and ⊕i≥0HomY (OY , ωiY ). When ωX andωY are ample, this gives rise to an algebraic isomorphism

X ' Proj(⊕i≥0HomX(OX , ωiX)) ∼→ Proj(⊕i≥0HomY (OY , ωiY )) ' Y .

When ωX and ωY are anti-ample, one proceeds in a similar way with the algebras⊕i≥0HomX(OX , ω−iX ) and ⊕i≥0HomY (OY , ω−iY ). It is worth saying that this proofdoes not makes use of Orlov’s representability theorem 2.15.

2.3.2 Characterization of Fourier-Mukai functors

We have already seen that a spanning class may be used to test if an exact fullyfaithful functor is an equivalence of categories. But one can also find a suitablespanning class for the derived category Db(X) and use it to state conditions foran integral functor to be a Fourier-Mukai functor.

The results in the first part of this section are valid in arbitrary characteristic,while starting from Proposition 2.56 we need to assume that the characteristic iszero.

The following two propositions are taken from [61].

Proposition 2.52. On a smooth proper variety X the skyscraper sheaves Ox forma spanning class for the derived category Db(X).

Proof. For everyM• ∈ Db(X) and every point x ∈ X there is a spectral sequenceEp,q2 = ExtpOX (H−q(M•),Ox) =⇒ Ep+q∞ = Homp+q

D(X)(M•,Ox). If M• 6= 0, then


there exists an integer q such that Hq(M•) 6= 0. Let q0 be the maximum of suchq’s and let x be a point in the support of Hq0(M•). Then we have a nonzeroelement e ∈ E0,−q0

2 = HomD(X)(Hq0(M•),Ox), and that element survives to givea nonzero element of E−q0∞ = Hom−q0D(X)(M

•,Ox). This implies that the skyscrapersheaves Ox form a spanning class on the right for the derived category Db(X),that is, they satisfy Condition 2 in Definition 2.1. Note that this does not requireX to be smooth, while this is necessary to prove Condition 1. Take thenM• 6= 0.Since X is smooth, ωX is a line bundle, so that N • =M• ⊗ ωX 6= 0 and we canapply the above argument to N • and find q0 and x such that Ext−q0OX (N •,Ox) =Hom−q0D(X)(N

•,Ox) 6= 0. By Serre duality Homn+q0D(X)(Ox,N

•) ' Ext−q0OX (N •,Ox)∗ 6=0, thus finishing the proof.

We also need to ascertain when the derived category Db(X) is indecompos-able.

Proposition 2.53. Let X be a smooth proper variety. Then Db(X) is indecompos-able if and only if X is connected.

Proof. If X is not connected, write X = X1

∐X2, and then Db(X) ' Db(X1) ⊕

Db(X2). Assume now that X is connected and that there exist full nontrivialsubcategories A1, A2 with Db(X) ' A1 ⊕ A2. For any integral closed subvarietyY → X, the sheaf OY is indecomposable, so that it is isomorphic to an objecteither in A1 or A2. Moreover, for every point y ∈ Y , the sheaf Oy is isomorphicto an object in the same Aj as OY , because otherwise HomD(X)(OY ,Oy) = 0 andthis is not true. Let Xj be the union of all integral subvarieties Y such that OY isisomorphic to an object of A2. Then X1, X2 are closed subsets and X = X1

∐X2,

because if y ∈ X1 ∩ X2, then Oy is isomorphic both to an object of A1 and anobject of A2, and this is absurd. Since X is connected, one of the Xj ’s, say X2, isempty. Then for every object K• in Db(X2) one has

HomiD(X)(K•,Ox) = 0 , for any i ∈ Z, x ∈ X

and therefore K• ' 0 because the skyscraper sheaves form a spanning class byProposition 2.52.

Let X be a smooth projective variety.

Definition 2.54. A sheaf F on X is special if F ⊗ωX ' F . An object F• of Db(X)is special if F• ⊗ ωX ' F• in Db(X). 4

Then, when the canonical bundle ωX is trivial, every object of Db(X) isspecial. An object F• of Db(X) is special if and only if its cohomology sheavesHi(F•) are special sheaves, as the following proposition shows.


Proposition 2.55. Let F• be a complex in Db(X) such that all its cohomologysheaves Hi(F•) are special sheaves. If f i : Hi(F•) → Hi(F•) ⊗ ωX are isomor-phisms of sheaves, there is an isomorphism f : F• → F• ⊗ ωX in the derivedcategory such that Hi(f) = f i for every i.

Proof. We proceed by induction on the number of nonzero cohomology sheaves.If Hn(F•) is the highest nonzero cohomology sheaf, we can assume that Fm = 0for m > n. Assume that F• has only a nonzero cohomology sheaf. Then F• 'Hm(F•)[m] in Db(X) and we set f = fm[m] : F• → F• ⊗ ωX . In the generalcase, we can assume by induction that there is an isomorphism f : F•≤(n−1) →F•≤(n−1)⊗ωX in the derived category inducing in cohomology the morphisms f i

for i ≤ n− 1. Consider the exact triangle in K(Qco(X))

F•≤(n−1)i−→ F• → Cone(i)

β−→ F•≤(n−1)[1] .

Note that β is homotopic to zero, so β = 0 in K(Qco(X)) and then also in thederived category. Since Cone(i) ' Hm(F•)[m] in the derived category, we have acommutative diagram in Db(X) whose arrows are exact triangles

F•≤(n−1)i //

f ′'

F• // Cone(i)β=0 //

fm[m]'

F•≤(n−1)[1]

f ′[1]'

F•≤(n−1)i // F• // Cone(i)

β=0// F•≤(n−1)[1] .

Then, there is an isomorphism f : F• → F• ⊗ ωX in Db(X) which completes thediagram.

From this point on, we assume that the base field k has characteristic zero.

Proposition 2.56. [202, 61] Let X and Y be smooth projective varieties of the samedimension n, and let K• be a kernel in Db(X × Y ). The following conditions areequivalent:

1. ΦK•

X→Y is a Fourier-Mukai functor;

2. ΦK•

X→Y is fully faithful and Lj∗xK• is a special object of Db(Y ) for all x ∈ X.

In particular, if Q is a sheaf on X × Y strongly simple over X, then ΦQX→Y is aFourier-Mukai transform if and only if Qx is special for all x ∈ X.

Proof. We can assume that Y is connected so that Db(Y ) is indecomposable byProposition 2.53. To prove that 2 implies 1, in view of Propositions 2.52 and 2.5 we


need to show that ΦK•

X→Y SX(Ox) ' SY ΦK•

X→Y (Ox) for every closed point x. Indeed,by the speciality of the complexes Lj∗xK•, we have

ΦK•

X→Y SX(Ox) ' Lj∗xK•[n] ' Lj∗xK•ωY [n] ' SY ΦK•

X→Y (Ox) .

The fact that 1 implies 2 is proved similarly.

The results of the previous proposition will be mostly used in the followingform.

Proposition 2.57. Assume that X and Y are smooth projective varieties of the samedimension with trivial canonical bundles and that K• is an object in Db(X × Y )strongly simple over X. Then the functor ΦK

•

X→Y is a Fourier-Mukai functor andthe functor ΦK

•∨[n]Y→X is a quasi-inverse to ΦK

•

X→Y . In particular, if Q is a locally freesheaf on X × Y strongly simple over X, the functor ΦQ

∗[n]Y→X is a quasi-inverse for

ΦQX→Y .

Corollary 2.58. Let X, Y be as in Proposition 2.57 and let K• be an object inDb(X × Y ) which is strongly simple over X. Then K• is strongly simple over Yas well and ΦK

•

Y→X is a Fourier-Mukai functor with inverse ΦK•∨[n]

X→Y .

Proof. By Proposition 2.57, ΦK•∨[n]

Y→X is an exact equivalence, and then K•∨ isstrongly simple over Y by Theorem 1.27. By Remark 1.26, K• is strongly sim-ple over Y , so that the statement follows again from Proposition 2.57.

We are now in a position to show that the composition of two Fourier-Mukaitransforms may fail to be a Fourier-Mukai transform.

Example 2.59. Let X be a K3 surface. We shall give an introduction to the geom-etry of K3 surfaces in Chapter 4; what we shall need here is that ωX ' OX andH1(X,OX) = 0. Let us consider the integral functor Φ = ΦI∆

X→X : Db(X)→ Db(X),where I∆ is the ideal sheaf of the diagonal in X ×X. One easily checks that I∆

is strongly simple, so that Φ is a Fourier-Mukai functor by Corollary 2.58. Againa straightforward computation shows that Φ(OX) ' OX [−2]. Moreover, if L is aline bundle on X which has no cohomology in every degree, one has Φ(L) ' L[−1].Comparing the two results, we see that for k big enough, the kernel of iteratedcomposition Φk is not a shifted sheaf. 4

Let X and Y be smooth projective varieties and K• a kernel in Db(X × Y ).We consider another pair X, Y of smooth projective varieties and another kernel

K• in Db(X × Y ). We then have a kernel K•L K• in Db(X × X × Y × Y ).

Lemma 1.28 about the product of integral functors can be strengthened inthe case of Fourier-Mukai functors.


Corollary 2.60. If the integral functors ΦK•

X→Y and ΦK•

X→Y are Fourier-Mukai func-

tors, then the functor ΦK•

L

K•X×X→Y×Y is a Fourier-Mukai functor as well.

Proof. In view of Lemma 1.28 and Proposition 2.56 we need only to show that for

evey closed point x, x) ∈ X×X the restriction Lj∗(x,x)(K•

L K•) is a special object.

From one side, one has the isomorphism Lj∗(x,x)(K•

L K•) ' Lj∗xK•

LLj∗xK•. From

the other, as ωY×Y ' ωY ωY , we have

Lj∗(x,x)(K•

L K•)⊗ ωY×Y ' (Lj∗xK• ⊗ ωY )

L (Lj∗xK• ⊗ ωY ) .

In a number of important examples that will be thoroughly investigated inthe next chapters, the hypotheses of Proposition 2.56 are met when Y is a con-nected component of the moduli spaces of simple sheaves on X, X is a connectedcomponent of the moduli spaces of simple sheaves on Y , andQ is the correspondingbi-universal family (provided it exists). In the case of surfaces, there are particularresults that will be very useful. We describe here some of them.

Let Y be a smooth projective surface and X a fine moduli space of specialstable sheaves on Y with fixed Mukai vector v (cf. Eq. (1.1)). Let Q be a universalsheaf on X × Y for the corresponding moduli problem, so that Q is flat over Xand Qx is a stable special sheaf on Y with Mukai vector v. Given closed points xand z in X, one has χ(Qx,Qz) = −v2 by Equation (1.7).

The following result can be found in [70].

Proposition 2.61. Assume that X is a projective surface.

1. X is smooth if and only if v2 = 0.

2. In this case, the integral functor ΦQX→Y : Db(X)→ Db(Y ) is a Fourier-Mukaifunctor.

Proof. Let x be a closed point of X. Since the Hom1X(Qx,Qx) is the tangent space

at x to the moduli space X of the sheaves Qx on Y , one has that X is smooth atx if and only if dim Hom1

X(Qx,Qx) = 2. To compute this dimension, we note thatthe sheaf Qx is stable and special, so that HomY (Qx,Qz) ' k by the stabilityand Hom2

X(Qx,Qx) ' k by Serre duality. It follows that −v2 = χ(Qx,Qx) =2− dim Hom1

X(Qx,Qx) which proves the first claim.

Assume now that v2 = 0. If x and z are different closed points of X, onehas that HomY (Qx,Qz) = 0 because Qx and Qz are nonisomorphic stable sheaveswith the same Chern characters. Since the sheaves QX are special, Serre duality


gives Hom2X(Qx,Qz) = 0. Finally, from χ(Qx,Qz) = −v2 = 0 we deduce that also

Hom1X(Qx,Qz) = 0, so that Q is strongly simple over X. The claim follows now

from Proposition 2.56.

Remark 2.62. Proposition 2.61 holds also true for pure stable sheaves in the senseof Simpson as described in Section C.2, because pure stable sheaves have theproperties of torsion-free stable sheaves we have used in its proof. 4

2.3.3 Fourier-Mukai functors between moduli spaces

We would like to show that in many cases, integral transforms define in a naturalway algebraic morphisms between moduli spaces.

Let Φ: Db(X)→ Db(Y ) be an integral functor, where X and Y are smoothprojective varieties. Let MX,P be the functor associating to any variety T theset of equivalence classes of all coherent sheaves E on T × X, flat over T andwhose restrictions Et = j∗t E to the fibers Xt ' X of πT : T × X → T haveHilbert polynomial P . Here, two sheaves E and E ′ are considered to be equivalentif E ' E ′⊗π∗TL for a line bundle L on T . Furthermore, let MX be a subfunctor ofMX,P parameterizing WITi sheaves for a certain index i. Corollary 1.9 implies thatif E is in MX(T ), the sheaves E = ΦiT (E) are flat over T , so that for a fixed i thefibers (E)t ' Et have the same Hilbert polynomial P . Moreover ΦiT (E ⊗ π∗TL) 'ΦiT (E) ⊗ π∗TL. Thus, ΦiT maps MX(T ) to MY,P (T ). The polynomial P can becomputed by the Grothendieck-Riemann-Roch formula in terms of P , the Chernclasses of the kernel K• of Φ and the Todd class of X.

By compatibility of integral functors with base change (Proposition 1.8), Φi

induces a morphism of functors

ΦiM : MX →MY,P .

Proposition 2.63. Assume that MX has a coarse moduli scheme MX .

1. If there is a subfunctor MY ⊂MY,P containing the image of ΦiM that is alsocoarsely representable by a moduli scheme MY , then the integral functor Φgives rise to an algebraic morphism of schemes

ΦiM : MX →MY .

2. If Φ is a Fourier-Mukai functor, then the image functor MY = Φi(MX)is coarsely representable by a moduli scheme MY , and Φ induces a schemeisomorphism

ΦiM : MX∼→ MY .

Moreover MX is a fine moduli scheme (that is, it represents the modulifunctor MX) if and only if MY is a fine moduli scheme.


Proof. 1. Since MY is coarsely represented by MY , there exists a morphism offunctors MY → Hom(−,MY ), where the latter is the functor of points of MY .The composition with Φi is a morphism of functors MX → Hom(−,MY ) which,by the definition of coarse moduli, factors in a unique way through a morphism offunctors Hom(−,MX) → Hom(−,MY ). This corresponds to a scheme morphismΦi : MX → MY . Part 2 is straightforward, due to the uniqueness of the coarsemoduli of a functor.

An important example is given by the moduli functor of skyscraper sheavesOx on X. Assume that the skyscraper sheaves are all WITi; this happens forinstance if the kernel of Φ is a concentrated complex, in which case they areWIT0. Then we have:

Corollary 2.64. If Φ is a Fourier-Mukai functor, then X is a fine moduli space forthe moduli functor MY of the sheaves Φi(Ox) over Y .

To illustrate another example, let X and Y be polarized smooth projectivevarieties (see Section C.2), and let Mss

X,P , MssY,P

be the corresponding modulifunctors of (Gieseker) semistable sheaves. Assume that all semistable sheaves Fin Mss

X,P are WITi and that their images Φi(F) are semistable. We have:

Corollary 2.65. 1. Φi induces a morphism of schemes ΦiM : MssX,P →Mss

Y,P.

2. If Φ is a Fourier-Mukai functor and Φi(MssX,P ) = Mss

Y,P, the induced mor-

phism is an isomorphism of schemes ΦiM : MssX,P∼→ Mss

Y,P.

Corollary 2.65 implies that Φi transforms S-equivalent semistable sheaveson X to S-equivalent sheaves on Y (for the notion of S-equivalence, see SectionC.2). However, Φi may transform non-S-equivalent semistable sheaves on X toS-equivalent sheaves on Y , even if Φ is a Fourier-Mukai functor. Thus, in generalthe morphism ΦiM : Mss

X,P →MssY,P

induced by a a Fourier-Mukai functor may failto be injective or surjective. There is however a partial result.

Corollary 2.66. If Φ is a Fourier-Mukai functor and there is a stable sheaf Fin Ms

X,P such that Φi(F) is stable, the functor Φi induces a surjective birationalmorphism

ΦiM : MX → MY ,

where MX and MY are the irreducible components of MssX,P and Mss

Y,Pwhich

contain [F ] and [Φi(F)], respectively.

Proof. If G is a semistable sheaf on X and ΦiM ([G]) = [Φi(F)], then Φi(G) is S-equivalent to F , and thus Φi(G) ' Φi(F) because Φi(F) is stable. Hence, G ' F by


the invertibility of Φ, and the fiber of ΦiM over the point [Φi(F)] is a single point.Since Mss

X,P and MssY,P

are projective (see Theorem C.6), by Zariski’s main theorem[136, 4.4.3] there exist open neighborhoods V of [Φi(F)] and U = (ΦiM )−1(V ) of[F ] such that ΦiM : U ∼→ V . Moreover, ΦiM (M) = MY because ΦiM (M) is irreducibleand contains [Φi(F)]. Then ΦiM : MX → MY is birational.


Historical remarks. As we already mentioned, the first instance of a Fourier-Mukaitransform is contained in Mukai’s 1981 paper [224]. A Fourier-Mukai transformon K3 surfaces was first constructed by the authors of this book in 1994 [24] (seealso [26]). Later a similar construction was done by Mukai [228]. We shall studyFourier-Mukai transforms on K3 surfaces in Chapter 4. Other examples of Fourier-Mukai transforms will be described in Chapter 6 as relative integral functors.

The talk by Bondal and Orlov at the 2002 International Congress of Math-ematicians [50] is a nice review of work done by them and others on equivalencesbetween derived categories of coherent sheaves.

Spherical objects. Fourier-Mukai functors can be constructed by using the so-called spherical objects, first introduced by Bondal and Polishchuk [51]. A complexE• in Db(X) is spherical if: (i) Homi

Db(X)(E•, E•) is equal to k for i = 0,dimX andto zero otherwise; (ii) E• is special, i.e., E•⊗ωX ∼→ E•. For instance, the structuresheaf of a Calabi-Yau variety is a spherical object. We exploited this property inExample 2.59, which indeed generalizes to any Calabi-Yau variety.

For any object E• of Db(X), one defines the twist functor TE• as the integral

functor whose kernel is the cone of the evaluation morphism E•∨L E• → O∆.

Whenever E• is spherical, the twist functor TE• is a Fourier-Mukai functor, asproved by Seidel and Thomas [265].

Results for singular varieties. Kawamata proved a generalization of Orlov’s rep-resentability theorem 2.15 to stacks associated with normal varieties with quo-tient singularities [176]. A characterization of Fourier-Mukai functors on Cohen-Macaulay varieties was given in [144, 143].

An alternative setting: differential graded categories. Though they are a powerfultool in algebraic geometry as well as in algebraic analysis, representation theoryand several other branches of mathematics, derived categories suffer from a numberof drawbacks. In particular, the underlying triangulated structure appears too poorto allow for an entirely satisfactory description of functors between these categoriesand of natural algebraic or homotopical operations. Bondal and Kapranov [46]proposed the idea of using differential graded categories as an “enhancement” ofderived categories in order to provide a more flexible and rich environment.

2.4. Notes and further reading 79

Differential graded categories — whose first appearance in the literature datesback to the 1960s [179] — can be thought of as “differential graded algebras withmany objects,” pretty much in the same vein as additive categories can be thoughtof as “rings with many objects.” The basics of differential graded categories arebriefly presented in Section A.4.4. For a more detailed overview the reader isreferred to Keller’s beautiful exposition [178]. Here we shall limit ourselves tofocus attention on a few issues that appear to be more relevant to our purposes.

The category dgcatk of small differential graded k-categories (Definition A.50)admits a structure of model category, whose weak equivalences are the quasi-equivalences (Theorem A.51). One denotes byHo(dgcatk) the localization of dgcatkwith respect to quasi-equivalences.

As proved in [105, 284], the monoidal category (Ho(dgcatk),L⊗ ) admits an

internal Hom-functor RHom (see Theorem A.56). Within this framework, Toen[284] has recently worked out a version of derived Morita theory, where the mor-phisms between dg-categories of modules over two dg-categories E, F are describedas the dg-category of (E-F)-bimodules.

As an application of his theory, Toen proved some results that can be viewedas a strengthening of Orlov’s representability theorem 2.15. Let us consider theAbelian category Qco(X) of quasi-coherent sheaves on an algebraic variety X. Weshall denote by Ddg(X) the dg-derived category Ddg(Cdg(Qco(X))) (see DefinitionA.54); one has that the homotopy category H0(Ddg(X)) is equivalent to D(X).As Theorem A.57 shows, given two algebraic varieties X, Y , there is naturalisomorphism in Ho(dgcatk)

Ddg(X × Y ) ' RHomc(Ddg(X), Ddg(Y )) ,

where RHomc denotes the full subcategory of RHom consisting of coproductpreserving quasi-functors. In particular, when X and Y are smooth and projective,it turns out (Equation A.10) that

parfdg(X ×k Y ) ' RHom(parfdg(X),parfdg(Y )) ,

where parfdg(X) is the full sub-dg-category of Ddg(X) whose objects are the per-fect complexes. We can rephrase this equivalence by saying that, in the dg envi-ronment, all functors are integral functors.

One should compare these results with the representability theorem [47, 6.8]proved by Bondal, Larsen, and Lunts. Actually, the dg-category parfdg(X) can beviewed as a standard enhancement of the derived category D(X) in the sense ofDefinition [47, 5.1].

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Chapter 3

Fourier-Mukai on Abelian

varieties

Introduction

Mukai’s 1981 paper [224] contains, in one way or another, in a more or less explicitform, many of the ideas that have been introduced and developed in subsequentyears in connection with Fourier-Mukai transforms. Thus these ideas are oftenat the core of the theory of integral functors that we have quite systematicallydeveloped in the first chapter of this book. As a result, we can prove most of theresults presented in this chapter quite straightforwardly. So, while the treatmentof this chapter is fairly close in spirit to the original paper by Mukai [224], thedetails of the arguments are often rather different.

In this chapter we review very briefly the basic definitions concerning Abelianvarieties, the dual Abelian variety, and the Poincare bundle P (later in the chapterwe shall also discuss the notion of polarization of an Abelian variety). We thenintroduce the integral functor associated with the kernel P and show immediatelythat it is a Fourier-Mukai transform. We study some properties of this transformand apply it to study some classes of sheaves on Abelian varieties (unipotent andhomogeneous bundles, Picard sheaves).

The chapter ends with the description of a property of Fourier-Mukai trans-forms that we encounter here for the first time: under suitable conditions, it pre-serves the stability of the sheaves it acts on. We shall meet this property again inother situations, e.g., for Fourier-Mukai transforms on K3 surfaces and for relativeFourier-Mukai transforms on elliptic fibrations.


In Chapter 6, devoted to relative integral functors, we shall discuss a Fourier-


82 Chapter 3. Fourier-Mukai on Abelian varieties

Mukai transform on Abelian schemes (an example again due to Mukai [226]) whichis a quite straightforward generalization of the transform studied in this chapter.

A good deal of information about Fourier-Mukai transforms on Abelian va-rieties may be found in the book by Polishchuk [251], where several topics notconsidered here are covered, such as Theta functions, a proof of the Torelli theo-rem by means of the Fourier-Mukai transform, and others.

3.1 Abelian varieties

An Abelian variety X over a field k is a proper integral (i.e., irreducible andreduced) algebraic group, i.e., a proper integral variety for which there are mor-phisms

• mX : X ×X → X (the group law),

• ιX : X → X (the inverse morphism),

• e : Spec k→ X (the identity point),

such that usual relations are fulfilled:

1. (associativity) mX (mX × IdX) = mX (IdX ×mX);

2. (existence of the unit) mX (IdX × e) = m (e× IdX) = IdX ;

3. (property of the inverse) denoting by (IdX × ιX), (ιX × IdX) : G → G × Gthe morphisms (IdX × ιX)(x) = (x, ιX(x)) and (ιX × IdX)(x) = (ιX(x), x),one has mX (IdX × ιX) = mX (IdX × ιX) = e p, where p : G→ Spec kis the natural projection onto a point.

A homomorphism φ : X → Y between two Abelian varieties X, Y is a mor-phism which is also a group homomorphism. If φ is surjective and its kernel isfinite, it is called an isogeny. In this case one defines the exponent e(φ) as thesmallest positive integer such that e(φ)x = 0 for all x ∈ ker(φ).

As a matter of fact, the condition that X is a proper variety implies that thisgroup structure is commutative (which motivates the name of these varieties) [229,§4, Cor. 2]. So we shall use an additive notation and will write mX(x, x′) = x+x′

and ιX(x) = −x. The translation morphism τ is defined as τx(x′) = x + x′. Theidentity element will be denoted 0. We shall also use the notation µX(x, x′) =x− x′, and will denote by π1, π2 the projections onto the factors of X ×X.

Since the translation morphism τx is an isomorphism for every x ∈ X, andsmooth points in X exist, any Abelian variety is nonsingular.

3.1. Abelian varieties 83

We wish to show that any Abelian variety is projective. Given a line bundleL on X, we consider the line bundle on X ×X

Q = m∗XL ⊗ π∗1L−1 ⊗ π∗2L−1 .

It is a general fact that there exists a closed subvariety K(L) ⊂ X which is thelargest subscheme of X such that the restriction of Q to X×K(L) is trivial. K(L)is a subgroup of X (this follows from the so-called theorem of the square [229, §6,Cor. 4]), and its points x are characterized by the condition τ∗xL ' L. One has thefollowing result.

Lemma 3.1. If L is effective, then it is ample if and only if K(L) is finite.

Proof. Here we just sketch the proof; for details cf. [229, Ch. 2]. Let D be aneffective divisor such that L = OX(D) and define GD = x ∈ X | τ∗x (D) = D. IfK(L) is finite, GD ⊂ K(L) is finite as well. From this one proves that the linearsystem |2D| has no base points and defines a finite morphism into a projectivespace. By general theory this implies that L is ample [134, 2.6.1 or 4.4.2].

On the other hand, if L is ample, one proves that K(L) is finite. Indeed, ifthis is not true, let Y be the connected component of K(L) containing 0. Then Yis a nontrivial Abelian variety. One proves quite easily that L′ = L|Y ⊗(−1)∗Y (L|Y )is ample. On the other hand since Y ⊂ K(L), pulling back Q|Y×Y to Y by themorphism y 7→ (y,−y), one shows that the line bundle L′ is trivial, which is acontradiction.

By using this we may show that any Abelian variety X is projective, as forinstance is proved in [251, Thm. 8.12]. One can see that there exists an effectivedivisor D → X whose complement U in X is affine and contains the origin. LetY be the connected component of the origin in K(O(D)). Then the restriction ofO(D) to Y is trivial, so that D is not contained in Y , and it does not intersect it.This implies that Y is contained in the affine variety U , which in turn implies thatY , being proper, reduces to a point, so that K(O(D)) is finite. By the previouslemma, X is projective.

The dual Abelian variety X and the Poincare bundle P on X × X may beintroduced as the solution to the problem of representing the Picard functor. Thisis the functor which to any variety T associates the group of equivalence classesof line bundles on X × T , where two line bundles are identified when they areisomorphic up to tensoring by the pullback of a line bundle on T . It is a generalfact that when X is projective, this functor is representable [3] by an algebraicgroup Pic(X). The connected component of the latter containing the origin isdenoted by Pic0(X) and represents line bundles that are topologically equivalentto the trivial line bundle (i.e., line bundles whose first Chern class vanishes). In


the case at hand, where X is an Abelian variety, Pic0(X) is an Abelian variety aswell; it is usually denoted X and called the dual Abelian variety to X. The factthat X represents the Picard functor means that there is a universal line bundle Pon X × X, called the Poincare bundle. Universality means that given a variety Tand a line bundle L on X × T , whose restrictions to the fibers of pT : X × T → T

have vanishing first Chern class, there exists a unique morphism φ : T → X suchthat L ' (IdX × f)∗P ⊗ p∗TN , where N is a line bundle on T . Thus, if a pointξ ∈ X corresponds to a line bundle L on X, one has

Pξ = P|X×ξ ' L .

Analogously, we shall denote Px = P|x×X if x ∈ X.

The Poincare bundle can be normalized by the condition that P0 = P|0×Xis the trivial line bundle on X. If we consider P as a relative line bundle withrespect to the projection X × X → X, we obtain a morphism X → Pic0(X)

which is actually an isomorphim canX : X → ˆX.

Let us consider the line bundle m∗XL ⊗ π∗1L−1 on X × X; by the universalproperty of the Poincare bundle, there is a unique scheme morphism φL : X → X

such that(1× φL)∗P ' m∗XL ⊗ π∗1L−1 ⊗ π∗2N

for certain line bundle N on X, which is actually isomorphic to L−1 as a conse-quence of the normalization of the Poincare bundle. Thus,

(1× φL)∗P ' m∗XL ⊗ π∗1L−1 ⊗ π∗2L−1 . (3.1)

We then haveφL(x) = τ∗xL ⊗ L−1 .

One can also check that φL is actually a group morphism (this is again a conse-quence of the theorem of the square [229, §6, Cor. 4]), i.e., it is a homomorphismof Abelian varietes. The kernel kerφL is easily shown to coincide with the schemeK(L). The latter is a subgroup of X, hence it acts freely on X, and φL factorsthrough the quotient X → X/K(L).

3.2 The transform

Let X be an Abelian variety over k, whose dimension we denote by g, and let X beits dual Abelian variety. In the rest of the chapter we assume that k has character-istic zero. Mukai’s seminal idea [224] was to use the normalized Poincare bundleP to define an integral functor between the derived categories of coherent sheaveson X and X, which turns out to be an equivalence of triangulated categories.

3.2. The transform 85

Theorem 3.2. The integral functor ΦPX→X : Db(X) → Db(X) is a Fourier-Mukai

transform. The functor ΦP∗[g]

X→X is quasi-inverse to ΦPX→X.

Proof. The Poincare bundle is strongly simple over X (cf. Definition 1.30) andthe canonical bundle of an Abelian variety is trivial, so that by Corollary 2.58,which can be applied because k has characteristic zero, ΦP

X→X is a Fourier-Mukaifunctor.

It is therefore natural to give the following definitions:

Definition 3.3. The Abelian Fourier-Mukai transform is the functor

S = ΦPX→X : Db(X)→ Db(X) .

The dual Abelian Fourier-Mukai transform is the functor

S = ΦP∗

X→X : Db(X)→ Db(X) .

4

Mukai considers instead of S the functor S : Db(X) → Db(X) given by S =ΦPX→X . The relation between S and S is given by the isomorphism of functors

S S ' ι∗X [−g] .

This is easily proved as a consequence of the isomorphism (id× ιX)∗P ' P∗.Recall (Definition 1.6) that a coherent sheaf F on X is WITi if its Abelian

Fourier-Mukai transform reduces to a single coherent sheaf F located in degree i,that is S(F) ' F [−i], and that it is ITi if in addition F is locally free. By basechange theory, a coherent sheaf F is ITi if and only if Hj(X,F ⊗Pξ) = 0 for j 6= i

and every point ξ ∈ X.

The notions of WIT and IT sheaf also apply to the dual Fourier-Mukai trans-form S. A consequence of the invertibility of the Fourier-Mukai transform (seeProposition 2.34) is

Corollary 3.4. If F is WITi, then F is WITg−i. Moreover F ' F .

Moreover, the spectral sequence (2.35) assumes the form

Ep,q2 = Sp(Sq(E)) =⇒

E if p+ q = g

0 otherwise .

Corollary 3.5. For every sheaf E on X, the sheaf S0(E) is WITg, while the sheafSg(E) is WIT0 (and hence IT0 by Proposition 1.7).


Example 3.6. Since Ox is a skyscraper sheaf, one has Hj(X,Ox ⊗ Pξ) = 0 forj 6= 0 and every point ξ ∈ X. Then Ox is IT0 and Ox ' Px. By Corollary 3.4, Pxis WITg for every point x ∈ X and Px ' Ox; this is an example of a sheaf whichis WIT but not IT.

Reversing the roles of X and X, one has that Oξ is IT0 with Oξ ' P∗ξ ' P−ξwhere −ξ is the opposite of ξ for the group law in X. Then Pξ is WITg with P∗ξ 'O−ξ, in other words S(Pξ) ' O−ξ[−g]. In particular, S(OX) = S(P∗

0) ' O0[−g]

where 0 denotes the origin of the Abelian variety X. 4

Theorem 3.2, whose proof is very simple thanks to the machinery developedin Chapter 1, encodes some nontrivial information about the cohomology of thePoincare bundle. Indeed, the following result [229, §13, Cor. 1] can be easily de-duced from it.

Proposition 3.7. One has

Hi(X × X,P) =

0 if i 6= g

k if i = g .

Proof. We have seen that S(OX) ' O0[−g], or, in other terms,

RiπX∗P =

0 if i 6= g

O0 if i = g .

By using the Leray spectral sequence we obtain Hi(X×X,P) ' H0(X, RiπX∗P),whence the claim follows.

As a matter of fact, Theorem 3.2 and Proposition 3.7 are essentially equiva-lent, since Mukai uses the latter to prove the first.

Example 3.8 (Unipotent bundles). A locally free sheaf U on X is said to be unipo-tent if there is a filtration

0 = U0 ⊂ U1 ⊂ · · · ⊂ Un−1 ⊂ Un = U

such that the quotients Ui/Ui−1 are isomorphic to OX for every i ≥ 1. Applyingthe Abelian Fourier-Mukai transform to the sequences

0→ Ui−1 → Ui → OX → 0

(cf. Eq. (1.10)) we can prove by induction on i that Ui is WITg for every i andthat its Fourier-Mukai transform Ui is a skyscraper sheaf supported at the origin0 ∈ X.


Conversely, if C is a skyscraper sheaf of length n on X supported at theorigin, we have a filtration

0 = C0 ⊂ · · · ⊂ Cn−1 ⊂ Cn = C

with Ci/Ci−1∼→ O0 for every i. The exact sequence of Fourier-Mukai transforms

gives that C is IT0 and that U = C is a unipotent bundle of rank n. Thus theFourier-Mukai transform establishes a one-to-one correspondence between unipo-tent bundles of rank n a skyscraper sheaves of length n supported at the origin 0.As a consequence of Proposition 2.63, the moduli scheme of unipotent bundles ofa given degree is isomorphic to the moduli scheme of skyscraper sheaves of lengthn supported at the origin 0. 4

Mukai derived from the Parseval formula (Proposition 1.34) many interestingconsequences [224]. We list here some of them:

Proposition 3.9. If F is a WITi sheaf on X, then

Hj(X,F ⊗ Pξ) ' Extj+g−iX

(Oξ, F)

andExtjX(Ox,F) ' Hj−i(X, F ⊗ P−x)

for every x ∈ X, ξ ∈ X and j ≥ 0.

Proof. One has Hj(X,F ⊗ Pξ) ' ExtjX(P∗ξ ,F) ' ExtjX(P−ξ,F) as Pξ is locallyfree. By Example 3.6 and the Parseval formula, ExtjX(P−ξ,F) ' Extj+g−iX (Oξ, F).The second formula is similar.

Corollary 3.10. If F is an ITi sheaf on X, then the Euler characteristic of F andthe rank of F are related by

χ(X,F) = (−1)i rk(F) .

Proof. One has χ(X,F) =∑j(−1)jhj(X,F) =

∑j(−1)j dim Extj+g−i

X(O0, F).

Since F is locally free, Serre duality gives Extj+g−iX

(O0, F) ' Hi−j(X, F∗ ⊗O0).

This vanishes unless j = i, and in that case we have dimH0(X, F∗ ⊗ O0) =rk(F).

The topological invariants of the Abelian Fourier-Mukai transform can becomputed by means of the Grothendieck-Riemann-Roch theorem in terms of thefirst Chern class of the Poincare bundle. In this case the map f : A•(X) ⊗ Q →A•(X) ⊗ Q introduced in Section 1.2 (which according to Corollary 2.40 is anisomorphism) takes the form

f(α) = πX∗(ch(P) · π∗X(α)) .


More explicitly, the Chern character of the Fourier-Mukai transform of a complexF• in Db(X) is given by

chk(S(F•)) =1k!πX∗

[c1(P)k · chg−k(F•)

]= f(chg−k(F•)) . (3.2)

Note that c1(P) has Kunneth type (1, 1); in particular, by embedding End(A1(X))as a submodule of A2(X × X) it corresponds to the identity of A1(X). Since fidentifies Ag−k(X) ⊗ Q with Ak(X) ⊗ Q, one can say that the Fourier-Mukaitransform “flips” the Chern character of the complex it acts on.

The Fourier-Mukai transform exchanges tensor products by line bundles ofdegree zero with translations by the group law. This parallels the property of theFourier transforms which interchanges products with sums. To prove this, let usdenote by τx : X → X the translation τx(y) = x+y. Similarly we have translationsτξ : X → X and τ(x,ξ) : X × X → X × X.

The following property of the Poincare bundle is well known:

Lemma 3.11. One has

τ∗(x,0)P ' P ⊗ π∗

XPx

τ∗(0,ξ)P ' P ⊗ π∗XPξ

for all x ∈ X, ξ ∈ X. Moreover,

(mX × 1)∗P ' π∗13P ⊗ π∗23P ,

where πij are the projections of X ×X × X onto the ij-th components.

Proof. Since (τ∗(x,0)P)ξ ' Pξ for every point ξ in X, the universal property of P

implies that τ∗(x,0)P ' P⊗π∗

XN for some line bundle N on X. The latter is shown

to coincide with Px by the normalization property P0 ' OX . This proves the firstformula, the second is similar. For the third we use those formulas together withthe cube theorem [229] (this states that if X, Y are proper integral varieties, Zany variety, (x0, y0, z0) ∈ X × Y × Z, and L is a line bundle on X × Y × Z suchthat all restrictions of L on x0 × Y × Z, X × y0 × Z and X × Y × z0 aretrivial, then L is trivial).

Proposition 3.12. [224] There are isomorphisms

S τ∗x ' (⊗P−x) S , S (⊗Pξ) ' τ∗ξ S

of functors Db(X)→ Db(X). Similarly, there are isomorphisms

S τ∗ξ ' (⊗Pξ) S , S (⊗P−x) ' τ∗x S

of functors Db(X)→ Db(X).


Proof. Let M• be an object of Db(X). Then π∗Xτ∗xM• ∼→ τ∗

(x,0)π∗XM• so that

S(τ∗xM•) = RπX,∗(τ∗(x,0)

π∗XM• ⊗ P) ' RπX,∗τ∗(x,0)

(π∗XM• ⊗ τ∗(−x,0)

P)

' RπX,∗τ∗(x,0)

(π∗XM• ⊗ P ⊗ π∗XP−x) ' RπX,∗(π

∗XM• ⊗ P ⊗ π∗

XP−x)

' RπX,∗(π∗XM• ⊗ P)⊗ P−x = S(M•)⊗ P−x

where the third equality follows from Lemma 3.11, the fourth from πX τ(x,0) = πXand the projection formula and the fifth again by the projection formula. Reversingthe roles of X and X we now find S τ∗ξ ' (⊗P∗−ξ) S = (⊗Pξ) S. Then(⊗Pξ) ' S τ∗ξ S[g] by Theorem 3.2, and therefore S (⊗Pξ) ' S S τ∗ξ S[g] '[−g] τ∗ξ S[g] ' τ∗ξ S. The same reasoning yields the last formula.

The Pontrjagin product of two coherent sheaves E and F on X is the sheafE ? F = mX∗(π∗1E ⊗ π∗2F). This product has a derived functor

Db(X)×Db(X)R?−→ Db(X)

(E•,F•) 7→ E• R? F• = RmX∗(π∗1E•

L⊗π∗2F•)

Proposition 3.13. [224] The Abelian Fourier-Mukai transform intertwines the Pon-trjagin and the tensor product, that is:

S(E• R? F•) ' S(E•)

L⊗S(F•) S(E•

L⊗F•) ' S(E•) R

? S(F•)[g] .

Proof. We have only to prove the first isomorphism. We use the isomorphism(mX × 1)∗P ' π∗13P ⊗ π∗23P where πij are the projections of X × X × X ontothe ij-th components (Lemma 3.11). The formula is obtained by successive basechanges and using the identities πX (mX × 1) = πX π13, π1 π12 = πX π13

and π2 π12 = πX π23.

S(E• R? F•) ' RπX∗(π

∗X(RmX∗(π∗1E•

L⊗π∗2F•)⊗ P))

' RπX∗(R(mX × 1)∗(π∗12(π∗1E•L⊗π∗2F•))⊗ P)

' RπX∗R(mX × 1)∗(π∗13π∗XE•

L⊗π∗23π

∗XF• ⊗ (mX × 1)∗P)

' RπX∗Rπ13∗(π∗13π∗XE•

L⊗π∗23π

∗XF• ⊗ π∗13P ⊗ π∗23P)

' RπX∗(π∗XE• ⊗ P

L⊗Rπ13∗π

∗23(π∗XF• ⊗ P))

' RπX∗(π∗XE• ⊗ P

L⊗π∗

XRπX∗(π

∗XF• ⊗ P)) ' S(E•)

L⊗S(F•) .


3.3 Homogeneous bundles

A sheaf M on an Abelian variety is called homogeneous if it is invariant undertranslations, that is, τ∗xM ∼→M for every point x ∈ X. We can give the same def-inition for objects in Db(X), saying that an object M• in Db(X) is homogeneouswhen τ∗xM• 'M• in Db(X) for every x. Since translations are isomorphisms theycommute with homology, so that if M• is homogeneous, then all the cohomologysheaves Hi(M•) are homogeneous. Conversely, if the cohomology sheaves Hi(M•)are homogeneous, induction on the number of nonzero cohomology sheaves to-gether with the exact triangle

M•≤n−1 →M•

≤n → Hn(M•)[−n]→M•≤n−1[1]

(where Hn(M•) is the last cohomology sheaf) shows that the complex M• ishomogeneous.

Proposition 3.12 allows one to characterize homogeneous sheaves and moregenerically homogeneous objects of Db(X). We need a preliminary result, whichwe prove using the notion of determinant bundle already considered in Chapter 1.

Lemma 3.14. LetM• an object of Db(X). IfM•⊗Pξ 'M• for every ξ ∈ X thenrk(M•) = 0 and all the cohomology sheaves Hi(M•) are skyscraper sheaves. Inparticular a sheaf M on X verifies the condition M⊗Pξ ' M for every ξ ∈ Xif and only it is a skyscraper sheaf.

Proof. Clearly M⊗Pξ 'M for every ξ ∈ X if M is a skyscraper sheaf. For theremaining statements, taking determinants inM•⊗Pξ 'M• we have det(M•) 'det(M•) ⊗ Prk(M•)

ξ (see Eq. (2.36)) so that Prk(M•)ξ is trivial for every ξ ∈ X.

Since Prk(M•)ξ ' Prk(M•)ξ we have that the image of the morphism X → X given

by ξ 7→ rk(M•)ξ, reduces to the origin. This is absurd unless rk(M•) = 0. WhenM• reduces to a sheaf M, then M has to be a skyscraper. Since M• ⊗Pξ 'M•

implies Hi(M•) ⊗ Pξ ' Hi(M•), we then have that the sheaves Hi(M•) areskyscrapers as well.

Proposition 3.15. The Abelian Fourier-Mukai transform of a skyscraper sheaf Mon X is a homogeneous locally free sheaf on X. Conversely, a homogeneous sheaf Fon X is WITg and locally free and its Fourier-Mukai transform F is a skyscraper.

Proof. A skyscraper sheaf M on X is IT0 so that M is a locally free sheaf on X.SinceM⊗P−x 'M by Lemma 3.14, then τ∗xM 'M by Proposition 3.12, and thisproves that M is homogeneous and WITg. For the converse, if F is homogeneous,then S(F) ⊗ Px ' S(F) again by the same proposition, and therefore all thecohomology sheaves Si(F) = Hi(S(F)) are skyscrapers by Lemma 3.14. Then

3.4. Fourier-Mukai transform and the geometry of Abelian varieties 91

Sp(Sq(F)) = 0 for p > 0 and the spectral sequence Ep,q2 = Sp(Sq(F)) (whichconverges to Ep+q∞ = F for p + q = g and Ep+q∞ = 0 otherwise) degenerates,proving that Sq(F) = 0 for q 6= g.

Corollary 3.16. The Abelian Fourier-Mukai transform induces an equivalence be-tween the category of homogeneous sheaves on X and the category of skyscrapersheaves on X.

As an application of the above results, Mukai ([224]) gave the following char-acterization of the homogeneous sheaves.

Corollary 3.17. A sheaf F on X is homogeneous if and only if it is isomorphic to⊕ni=1 Ui ⊗ Pi, where the Ui’s are unipotent bundles and the Pi’s line bundles of

degree zero.

Proof. If F is homogeneous, then it is WITg and F is a skyscraper, F =M′1⊕· · ·⊕M′i where each M′i is supported at a point ξi ∈ X. We can write M′i = τ∗ξiMi

where Mi is supported at the origin, so that the invertibility of the Fourier-Mukai transform gives F =

⊕i S(τ∗ξiMi) '

⊕i S(Mi)⊗Pξi by Proposition 3.12.

Moreover Ui = S(Mi) is a unipotent bundle by Example 3.8. The converse issimilar.

3.4 Fourier-Mukai transform and the geometry

of Abelian varieties

We can give an alternative approach to some general properties of Abelian varietiesby means of the Abelian Fourier-Mukai transform.

3.4.1 Line bundles and homomorphisms of Abelian varieties

Let us recall that a line bundle L on X defines a flat homomorphism φL : X → X,whose kernel is K(L).

Proposition 3.18. One has

1. φL is constant (and then equal to the zero map) if and only if L ∈ X.

2. If L and N are line bundles on X, then φL⊗N = φL + φN .

3. φL = φN if and only if L and N are algebraically equivalent, that is, L ⊗N−1 ∈ X or c1(L) = c1(N ).


Proof. 1. φL is constant if and only if L is a homogeneous line bundle. By Corollary3.17, this amounts to saying that L ' Pξ for some ξ ∈ X, that is, L ∈ X.

2. One has

(1× φL⊗N )∗P ' m∗X(L ⊗N )⊗ π∗1L−1 ⊗ π∗1N−1 ⊗ π∗2L−1 ⊗ π∗2N−1

' (1× φL)∗P ⊗ (1× φN )∗P ' (1× (φL, φN ))∗(π∗13P ⊗ π∗23P)

where 1 × (φL, φN ) : X × X → X × X × X is the morphism given by (x, x′) 7→(x, φL(x), φN (x′)). Now we apply the formula (1×mX)∗P ' π∗13P⊗π∗23P obtainedby applying Lemma 3.11 to X. Thus

(1× φL⊗N )∗P ' (1× (φL, φN ))∗((1×mX)∗P) ' (1× (φL + φN ))∗P .

By the universal property of P we get φL⊗N = φL + φN .

3. It follows directly from 1. and 2.

We also recall that due to the fact that the tangent bundle to an Abelianvariety is trivial, the Grothendieck-Riemann-Roch formula for a line bundle takesthe well-known form

χ(X,L) =1g!c1(L)g .

It follows that χ(X,L−1) = (−1)gχ(X,L).

A line bundle L on X is said to be nondegenerate when the morphism φL isfinite. Note that in this case φL is a separable isogeny, and in particular is etale,cf. [229].

Proposition 3.19. Let L be a line bundle on X.

1. If L is nondegenerate, then j∗(L|K(L))[−g] ' L−1 R? L, where j : K(L) → X

is the immersion.

2. If L is nondegenerate, there exists an integer i = i(L) such that H`(X,L) = 0for ` 6= i. Moreover i(L−1) = g − i(L).

3. deg φL = χ(X,L)2. Moreover, χ(X,L) = 0 when L is degenerate.

4. If L is nondegenerate, then it is ITi with i = i(L) and L is locally free ofrank (−1)iχ(X,L). Moreover φ∗LL ' Hi(X,L)⊗k L−1.

5. If L is ample, then it is effective.

Proof. 1. Since L is nondegenerate the morphism φL is flat, and one has j∗OK(L) 'φ∗LO0 ' Lφ∗LO0. Example 3.6, together with base change and Equation (3.1), now


gives

j∗OK(L)[−g] ' φ∗LS(OX) ' Rπ2∗((1× φL)∗P)

' Rπ2∗(m∗XL ⊗ π∗1L−1)⊗ L−1

' Rπ2∗((π1,mX)∗(m∗XL ⊗ π∗1L−1))⊗ L−1

' RmX,∗(π∗1L−1 ⊗ π∗2L)⊗ L−1

whence the statement follows.

2. By the first part, and by the definition of the Pontrjagin product, we havethat Rim∗(π∗1L−1⊗π∗2L) = 0 for i 6= g and that Rgm∗(π∗1L−1⊗π∗2L) ' j∗(L|K(L)).Taking cohomology, we have

Hp(K(L),L|K(L)) ' Hp+g(X ×X,π∗1L−1 ⊗ π∗2L)

'⊕

j+i=p+g

Hj(X,L−1)⊗k Hi(X,L) . (3.3)

If L is nondegenerate, the first member vanishes for p > 0 because K(L) is finiteand we thus deduce the result.

3. If L is nondegenerate, so that K(L) is finite, the previous formula gives

deg φL = length(K(L)) = hi(X,L)hg−i(X,L−1)

= (−1)iχ(X,L)(−1)g−iχ(X,L−1) = χ(X,L)2 ,

where i is the index of L.

If L is degenerate, deg φL = 0 because φL is not finite and we have to provethat χ(X,L) = 0. Since K(L) is not finite, we can find for any integer n analgebraic subgroup G → K(L) of order n. Then 1×φL factors through X ×X →X ×X/G → X × X so that m∗XL ⊗ π∗2L−1 = (1 × φL)∗(P ⊗ π∗1L) is the inverseimage of a line bundle on X×X/G. Thus χ(X×X,m∗XL⊗π∗2L−1) is divisible by nand since n is arbitrary large, this implies that χ(X×X,m∗XL⊗π∗2L−1) = 0. Now,(µX , π2) : X×X → X×X is an isomorphism, and then χ(X×X,π∗1L⊗π∗2L−1) =χ(X × X, (µX , π2)∗(m∗XL ⊗ π∗2L−1)) = 0. Moreover χ(X × X,π∗1L ⊗ π∗2L−1) =χ(X,L)χ(X,L−1) = (−1)gχ(X,L)2, and then χ(X,L) = 0.

4. Proceeding as in the first part we have

φ∗LS(L) ' Rπ2∗(π∗1L ⊗ (1× φL)∗P)

' Rπ2∗(m∗XL ⊗ π∗2L−1)

' Rπ2,∗((mX , π2)∗(π∗1L ⊗ π∗2L−1))

' Rπ2,∗(π∗1L ⊗ π∗2L−1) ' RΓ(X,L)⊗k L−1

' Hi(X,L)⊗k L−1[−i]


where the last equality is due to the second part. Then φ∗L(Hj(S(L))) = 0 for j 6= i

and φ∗L(Hi(S(L))) is locally free. Since φL is flat and surjective, it is is faithfullyflat, so that Hj(S(L)) = 0 for j 6= i and Hi(S(L)) is locally free; thus L is ITi.Moreover, φ∗LL ' Hi(X,L)⊗k L−1.

5. Let n be an integer such that Ln is very ample. By Lemma 3.1, Ln isnondegenerate. Since φLn = [n] φL, the line bundle L is nondegenerate as well.By the corollary at page 159 of [229] one has i(L) = i(Ln) = 0, so that

h0(L) = χ(L) =1g!c1(L)g > 0 .

3.4.2 Polarizations

A class of algebraic equivalence of ample line bundles H = [L] (or of ample divisors[D]) is called a polarization. By Proposition 3.18, two line bundles are algebraicallyequivalent if and only if they define the same morphism φL : X → X. Then themorphism is actually associated to the class, and we denote it by φ[L].

A polarization [L] is said to be principal if φ[L] is an isomorphism of Abelianvarieties, φ[L] : X ∼→ X. We have seen that this is equivalent to either deg φL = 1 orto χ(X,L) = 1. An important example of a principally polarized Abelian variety isthe Jacobian J(C) of a smooth projective curve C, the scheme which parameterizesthe lines bundle on C having degree zero. In this case the principal polarization isgiven by the equivalence class of the so-called Θ divisor on J(C).

Given a polarization H = [L] on X, we may use the Fourier-Mukai transformto endow the dual variety X with a polarization.

Corollary 3.20. If L is an ample line bundle on X, the line bundle det(L)−1 onX is ample as well.

Proof. We have φ∗LL∗ ' H0(X,L)⊗kL. Now, the locally free sheaf H0(X,L)⊗kLis ample (for the definition and main properties of ample locally free sheaves see[138]). Since a locally free sheaf is ample if and only if its pullback under a finitesurjective morphism is ample, the locally free sheaf L∗ is ample. Then the linebundle det(L)−1 is ample as well.

So H = [det(L)−1] is a polarization for X. If one identifies X with ˆX by

canX , we may iterate the construction, obtaining a polarization ˆH on X.

Corollary 3.21.ˆH = (−1)∗X(H).


Proof. Let M = det−1(L). Denoting Φ = ΦP∗

X→X , by Equation (3.2) we have

ˆH = c1(Φ(M)) = (−1)∗X(c1(L)) = (−1)∗X(H) .

When we are given a principally polarized Abelian variety (X, [L]) we shallidentify X with X by the isomorphism φ[L]. Then X×X is identified with X× Xand under this identification we have

P ' m∗XL ⊗ π∗1L−1 ⊗ π∗2L−1 . (3.4)

Thus the Abelian Fourier-Mukai transform S and the dual Abelian Fourier-Mukaitransform S are autoequivalences of D(X).

3.4.3 Picard sheaves

In this section, C is a smooth projective curve of genus g ≥ 1. For any integer dwe denote by Jd the Picard scheme parameterizing degree d line bundles on C andby p : C × Jd → C, q : C × Jd → Jd the projections. The universal Poincare linebundle on C × Jd is denoted by Pd. It is determined up to twisting by pullbacksof line bundles on Jd; however if we fix a point x0 ∈ C, we can normalize Pd byimposing that Pd|x0×Jd ' OJd .

The Jacobian of C is the Abelian variety J(C) = J0. In this case, the normal-ized Poincare bundle P0 is just the restriction of the normalized Poincare bundleon J(C)×J(C) to C×J(C), where we identify J(C) with its dual via the principalpolarization given by the Θ divisor, and C is embedded into J(C) via the Abelmap.

All the Picard schemes are isomorphic in a noncanonical way. The choice ofa point x0 ∈ C gives isomorphisms

λd : Jd ∼→ J(C)

[L] 7→ [L ⊗OC(−dx0)] .(3.5)

These isomorphisms are induced by the line bundle Pd ⊗ p∗OC(−dx0). The nor-malization of the Poincare sheaves give

Pd ⊗ p∗OC(−dx0) ' (1× λd)∗P0 . (3.6)

We consider the integral functors Φd = ΦPdC→Jd : Db(C)→ Db(Jd).

Definition 3.22. The Picard sheaves are the cohomology sheaves

Ed,n = Φ0d(OC(nx0)) , Fd,n = Φ1

d(OC(nx0)) ,

of the transformed complex Φd(OC(nx0)). 4


Picard sheaves were introduced by Schwarzenberger [264] and have been thesubject of study by many authors in connection with the geometry of Abelianvarieties (cf. [224, 180, 181]).

By base change in the derived category we have isomorphisms

Φd(OC(nx0)) ' Φ(1×λd)∗P0⊗p∗OC(dx0)C→Jd (OC(nx0)) ' λ∗dΦ0(OC((n+ d)x0)) (3.7)

and thenEd,n ' λ∗d(E0,d+n) , Fd,n ' λ∗d(F0,d+n) . (3.8)

The sheaf p∗ωC ⊗ P∗d induces an isomorphism

θd : Jd ∼→ J2g−2−d

[L] 7→ [p∗ωC ⊗ L−1](3.9)

which givesp∗ωC ⊗ P∗d ' (1× θd)∗P2g−2−d (3.10)

due to the normalization. Again by base change in the derived category we haveisomorphisms

θ∗d(Φ2g−2−d(OC(nx0))) ' ΦP∗d⊗p

∗ωCC→Jd (OC(nx0)) ' ΦP

∗d

C→Jd(ωC ⊗OC(nx0)) . (3.11)

Proposition 3.23. There is an isomorphism

θ∗dΦ2g−2−d(OC(nx0))∨ ' λ∗dΦ0(OC((d− n)x0))[1]

where as usual ∨ denotes the dual in the derived category.

Proof. By Proposition 1.15

ΦP∗d

C→Jd(ωC ⊗OC(nx0))∨ ' Φd(SC(ω−1C ⊗OC(−nx0))) ' Φd(OC(−nx0)[1]) .

One then applies (3.7).

For any d > 0, the Abel morphism Symd C → Jd in degree d (where Symd C

is the symmetric product of d copies of C) may be identified with the projectivemorphism associated with a coherent OJd -module Md (cf. [3]). Moreover Md isunivocally characterized by the property

HomT (f∗Md,N ) ' HomC×T ((1× f)∗P∗d , q∗TN )

for every morphism f : T → Jd and every quasi-coherent sheaf N on T , whereqT : C × T → T is the projection.

The following result is easily proved [264].

3.5. Some applications of the Abelian Fourier-Mukai transform 97

Proposition 3.24. There is an isomorphism

Md ' Φ1d(ωC) ' θ∗dF0,2g−2−d .

where Φd = ΦP∗d

C→Jd .

Proof. Relative duality and base change give

HomDb(T )(f∗(Φd(ωC))[1],N ) ' HomDb(Jd)(Φd(ωC)[1], f∗(N ))

' HomDb(C×Jd)(p∗ωC ⊗ P∗d [1], q∗f∗(N )⊗ p∗ωC [1])

' HomDb(C×Jd)(P∗d , q∗f∗(N ))

' HomDb(C×Jd)(P∗d , (1× f)∗q∗T (N ))

' HomDb(C×T )((1× f)∗P∗d , q∗T (N )) .

There is a spectral sequence whose second term is

Ep,q2 = HompC×T (H−q(f∗(Φd(ωC))[1]),N )

converging to Ep,q∞ = Homp+qDb(T )

(f∗(Φd(ωC))[1],N ). Since Φ1d(ωC) is the highest

cohomology sheaf of Φd(ωC) one has Ep,q2 = 0 for q < 0, so that

HomC×T ((1× f)∗P∗d , q∗T (N )) ' HomT (f∗T (Φ1d(ωC)),N ) ,

which proves thatMd ' Φ1d(ωC). The second isomorphism follows from (3.11).

3.5 Some applications of the Abelian Fourier-Mukai

transform

We discuss in this section two applications of the Abelian Fourier-transform: aproof of Atiyah’s characterization of semistable bundles on elliptic curves, and thepreservation of stability in the case of Abelian surfaces.

3.5.1 Moduli of semistable sheaves on elliptic curves

The structure of µ-stable and µ-semistable vector bundles on an elliptic curve wasdetermined by Atiyah [15] in a paper which now is considered to be a classic. Hisresults were used by Tu [285] to study the moduli spaces of µ-semistable sheaves onan elliptic curve. It turns out that all those results can be obtained in a very simpleway as an application of the Abelian Fourier-Mukai transform on an elliptic curve.This approach was introduced in [251] and [142] but our treatment is somehowdifferent.


Let X be an elliptic curve, understood as an Abelian variety of dimension 1.In other words, X is a smooth curve of genus 1 with a fixed point x0 which wetake as the zero of the group law on X. The line bundle L = OX(x0) induces aprincipal polarization on X so that we can identify X with X by the isomorphismφ[L]. Under this identification the Poincare bundle takes the form

P ' OX×X(∆ι)⊗ π∗1OX(−x0)⊗ π∗2OX(−x0) , (3.12)

where ∆ι is the graph of the isomorphism ι : X → X which maps a point x tothe opposite point ι(x) = −x (cf. (3.4)). We can then consider both the AbelianFourier-Mukai transform S and the dual Abelian Fourier-Mukai transform S asautoequivalences of D(X).

Since X has dimension 1, an object E• of Db(X) has only two topologicalinvariants, namely, the rank n = ch0(E•) and the degree d = ch1(E•) (which isnaturally identified with an integer number), so that the Chern character of E•can be written as ch(E•) = (n, d).

Applying Equation (3.2) or computing directly by Grothendieck-Riemann-Roch from Equation (3.12), we have:

Proposition 3.25. If the Chern character of E• is (n, d), then the Chern characterof the Abelian Fourier-Mukai transform S(E•) is (d,−n).

When the object E• reduces to a single sheaf E , one has:

Corollary 3.26. If E is WITi for some i = 0, 1 and d 6= 0, then µ(E) = −1/µ(E).

Corollary 3.27. If E is WIT0, then d(E) ≥ 0, and d(E) = 0 if and only if E = 0.If E is WIT1, then d(E) ≤ 0.

The key point for the study of µ-semistable sheaves on an elliptic curveis the following result [251, Lemma 14.5], whose proof is based on the Harder-Narasimhan filtration.

Proposition 3.28. Any indecomposable torsion-free sheaf on an elliptic curve X issemistable.

Proof. Let E be a torsion-free sheaf on X and let 0 ⊂ E1 ⊂ · · · ⊂ En = E beits Harder-Narasimhan filtration (cf. [155, 1.3]). Then the quotient sheaves Gi =Ei/Ei−1 are µ-semistable with µ(Gi) > µ(Gi+1). It follows that HomX(Gi,Gi+1) = 0and then Serre duality implies Ext1

X(Gi+1,Gi) = 0. Thus the Harder-Narasimhanfiltration splits. If E is indecomposable, then it is semistable.

Corollary 3.29. Let E be a semistable sheaf of rank n and degree d on an ellipticcurve X.


1. If d < 0, then E is IT1 with respect to both S and S, and both transformsare semistable.

2. If d > 0, then E is IT0 with respect to both S and S, and both transformsare semistable.

3. If d 6= 0, then E is locally free.

4. If d = 0 and E is stable, then E is a line bundle. Thus, any semistable sheafof degree 0 is WIT1 and E is a skyscraper sheaf. Moreover a torsion-freesheaf of degree 0 is semistable if and only if it is a homogeneous bundle.

5. If d = 0, E is S-equivalent to a direct sum of degree 0 line bundles:

E ∼⊕i

L⊕nii ,∑i

ni = n .

Proof. 1. For every point ξ ∈ X one has H0(X, E ⊗ Pξ) ' HomX(P∗ξ , E) = 0since E is semistable of negative degree. By Proposition 1.7, E is IT1. To prove thesemistability part, we can assume that E is indecomposable; then E is indecompos-able as well, so that it is semistable by Proposition 3.28. An analogous argumentproves the statement for S.

2. By Serre duality, one has isomorphisms

H1(X, E ⊗ Pξ)∗ ' HomX(E ⊗ Pξ,OX) ' HomX(E ,P∗ξ ) .

As above, the latter group is zero since E is semistable of positive degree, and thenE is IT0. The semistability of E is proved as in the first part. The proof for S issimilar.

3. Notice that if d 6= 0, by parts 1 or 2 E is ITi with respect to S andE is semistable of nonzero degree, so that it is IT1−i with respect to S. ThenE ' S1−i(Si(E)) is locally free.

4. Since one has H0(X, E⊗Pξ) ' HomX(P∗ξ , E), if E is stable of degree 0, thenH0(X, E⊗Pξ) = 0 unless E ' P∗ξ . It follows that if E is not a line bundle, it is IT1;the transform E is locally free of rank 0 by Proposition 3.25 so that E = 0. By theinvertibility of S, one has E = 0. This proves the first part. Now if E is semistableof degree 0, it has a Jordan-Holder filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = E whosequotients Gi = Ei/Ei−1 are stable of degree 0. Then the sheaves Gi are line bundlesof degree 0, that is, Gi ' Pξi for a point ξi ∈ X. Since the sheaves Pξi are WIT1

and Pξi ' Oι(ξi) (cf. Example 3.6), we deduce that E is WIT1 and E is a skyscrapersheaf. By Proposition 3.15, E is a homogenous bundle. A similar argument provesthat E is also WIT1 with respect to S and that S1(E) is a skyscraper sheaf.

5. The argument used in part 4 proves the statement.


Let us denote by Cohssn,d(X) the full subcategory of the category Coh(X) ofcoherent sheaves on X whose objects are semistable sheaves of rank n and degreed. Unlike the category Cohssµ (X) of semistable sheaves with slope µ, the categoryCohssn,d(X) is not additive. However, this category will be useful to our presentpurpose, which is to prove Atiyah’s results on the characterization of semistablesheaves on X. We denote by Skyn(X) the category of skyscraper sheaves on X oflength n.

Corollary 3.29 implies directly the following result.

Proposition 3.30. The Abelian Fourier-Mukai transform S induces equivalences ofcategories

Cohssn,d(X) ' Cohssd,−n(X) , if d > 0

Cohssn,0(X) ' Skyn(X) .

Moreover, the twist by L = OX(x0), which is a Fourier-Mukai transformΨ = Φδ∗LX→X : Db(X) ' Db(X), also induces an equivalence of categories

Cohssn,d(X) ' Cohssn,d+n(X) .

By appropriately composing the integral functors S and Ψ we obtain the followingresult.

Proposition 3.31. For every pair (n, d) of integers (n > 0), there is a Fourier-Mukai functor Φ : Db(X) ' Db(X) which induces an equivalence of categories

Cohssn,d(X) ' Cohssn,0(X) ,

where n = gcd(n, d). The integral functor Φ = S Φ induces then an equivalenceof categories

Cohssn,d(X) ' Skyn(X) .

Proof. If d = 0, we just take Φ = S (cf. Proposition 3.30). Assume now that d > 0.If n ≤ d, we reproduce the method which computes the greatest common divisorn of (n, d); there is a sequence of Euclidean divisions

d = a0n+ d1 with d1 < n

n = a1d1 + d2 with d2 < d1

...ds−2 = as−1ds−1 + n with n < ds−1

ds−1 = asn .

If we denote bj = (−1)j−1aj , the composition Φ = ΨbsSΨbs−1S · · ·Ψb1SΨa0

is the required integral functor. If n > d, we just start from the second step bytaking d1 = d, that is, Φ = Ψbs S Ψbs−1 S · · · Ψb1 S. Finally, if d < 0, byapplying S we reduce to the case (−d, n) and we can then proceed as above.


Remark 3.32. Via Proposition 3.31, the multiplicative properties of the Poincarebundle can be used to deduce Atiyah’s multiplicative structure of Cohssn,d(X), see[142]. 4

Corollary 3.33. Let E be a torsion-free sheaf of rank n and degree d on X. Thefollowing conditions are equivalent:

1. E is stable;

2. E is simple;

3. E is semistable and gcd(n, d) = 1.

Thus, the integral functors of Proposition 3.31 map stable sheaves to stable sheaves.

Proof. If E is stable, obviously it is simple. If E is simple, it is indecomposable, sothat it is semistable by Proposition 3.28. By Proposition 1.34, Φ(E) is simple aswell. Since Φ(E) is a skyscraper sheaf of length n = gcd(n, d), one has n = 1.

If E is semistable and gcd(n, d) = 1, then E is evidently stable.

We can also derive the structure of the coarse moduli space Mss(n, d) ofsemistable sheaves of rank n and degree d on X. Let us denote by SymnX then-th symmetric product of X.

Corollary 3.34. For every pair (n, d) of integers (n > 0), there is a Fourier-Mukaifunctor Φ : Db(X) ∼→ Db(X) which induces an isomorphism of moduli spaces

Mss(n, d) 'Mss(n, 0) ,

where n = gcd(n, d). The integral functor Φ = S Φ induces an isomorphism

Mss(n, d) ' SymnX .

Proof. The first part follows directly from Propositions 3.31 and 2.63. To provethe second part we need to show that the symmetric product SymnX is a coarsemoduli space for the moduli functor of skyscraper sheaves of length n onX. Indeed,once this result is established, we may use Propositions 3.31 and 2.63 to get theclaim.

We note that the moduli functor of skyscraper sheaves of length n on X

coincides with the moduli functor MssX,n of Simpson semistable sheaves with con-

stant Hilbert polynomial P (m) = n (cf. Section C.2). Then, this functor has acoarse moduli space Mss

X,n whose closed points are S-equivalence classes. Sincethe only simple skyscraper sheaves are the structure sheaves of the closed points,any skyscraper sheaf is S-equivalent to a direct sum ⊕iOnixi (with n =

∑i ni), so


that closed points of MssX,n are in a one-to-one correspondence with closed points

of SymnX. Though it is a standard result, for completeness we prove that thiscorrespondence is induced by an algebraic isomorphism.

First, we recall that the Hilbert-Chow morphism Hilbn(X)→ SymnX, whichmaps a zero-cycle of length n to its support, is an isomorphism, since X is a smoothcurve. Secondly, if T is a scheme and F is a sheaf on X × T , flat over T and suchthat Ft is a skyscraper sheaf of length n on Xt for every t ∈ t, then the modifiedsupport Supp0(F) (see Definition C.9) is a subscheme of X × T flat of degree nover T , that is, a T -valued point of the Hilbert scheme Hilbn(X). We then have afunctor morphism

MssX,n → Hom( • ,Hilbn(X)) ' Hom( • ,SymnX)

and thus, an algebraic morphism MssX,n → SymnX. We know that this morphism

is one-to-one on closed points; since SymnX is smooth, the morphism is an iso-morphism by Zariski’s main theorem [141, 11.4].

We shall consider similar properties for relatively semistable sheaves on anelliptic fibration in Section 6.4.

3.5.2 Preservation of stability for Abelian surfaces

We analyze in this section a first instance of an important feature of the Fourier-Mukai transforms for Abelian surfaces, i.e., that in suitable circumstances theypreserve the stability of sheaves. We follow mainly [203] (cf. also [108]).

The main result is the following. Let X be an Abelian surface with a fixedpolarization H. We consider the dual Abelian surface X with the dual polarizationH we have defined in Section 3.4.2.

Theorem 3.35. Let E be a µ-stable locally free sheaf on X with µ(E) = 0 which isnot a flat line bundle (i.e., it is not a line bundle with vanishing first Chern class).Then E is IT1, and its Fourier-Mukai transform E is µ-stable with respect to thedual polarization H.

The key result in this context is the following lemma.

Lemma 3.36. If E is an IT0 sheaf on X then deg(E) ≥ 0, with equality if and onlyif E is a skyscraper.

Proof. Let T be the torsion subsheaf of E , and F = E/T . Then µ(F) ≤ µ(E) withequality if and only if T is a skyscraper (or it is zero).


Let us assume that F is nonzero and µ(F) ≤ 0. We have S2(T ) = 0 so thatapplying the functor S to the exact sequence

0→ T → E → F → 0

and using that E is IT0, we obtain that F is IT0 as well. Thus, F = S0(F) islocally free, WIT2 (cf. Corollary 3.4) and satisfies µ(F) ≥ 0.

Since H2(X, F) = F ⊗O0 6= 0, there is a nonzero morphism F → OX , henceF is not µ-stable. Let

0→ K → F → G → 0

be a destabilizing sequence, where K is µ-stable, so that µ(K) ≥ µ(F) ≥ 0. ByProposition 1.15, one has S(F∗) ' F∨ = RHomOX (F ,OX), whence by takingcohomology and taking into account that F is torsion-free, we obtain

S2(F∗) ' Ext2OX (F ,OX) = 0

so that H0(X, F ⊗ Px) ' H2(X, F∗ ⊗ P−x) = 0 for all x ∈ X. Then H0(X,K ⊗Px) = 0 for all x, so that K is not a flat line bundle. Moreover H2(X,K⊗Px) = 0as K is µ-stable, so that K is IT1.

On the other hand we have S0(G) ' S1(K), where S1(K) is WIT1 by Corol-lary 3.4, while S0(G) is WIT2 by Corollary 3.5. This is a contradiction, so thateither F = 0 or µ(F) > 0. In the first case, E has nonnegative degree and has degreezero if and only if it is a skyscraper. In the second case, we have deg(E) > 0.

Corollary 3.37. If E is a WIT2 sheaf on X then deg(E) ≤ 0, with equality if andonly if E is a homogeneous bundle.

Proof. This follows from the previous lemma and Proposition 3.15.

Proposition 3.38. If E is a µ-semistable WIT1 sheaf on X with µ(E) = 0, then Eis µ-semistable.

Proof. Let us assume that E is not µ-semistable, and let 0 → F → E → G → 0be a destabilizing sequence; possibly by modding its torsion out, we may assumethat G is torsion-free. Applying the functor S to this sequence we obtain

0→ S0(G)→ S1(F)→ E → S1G → S2(F)→ 0 (3.13)

together with S0(F) = S2(G) = 0. By Corollary 3.5, the sheaf S0(G) is WIT2 andhence by Corollary 3.37 it satisfies deg(S0(G)) ≤ 0 with equality if and only ifS0(G) is a homogeneous bundle, and analogously deg(S2(F)) ≥ 0 with equality ifand only if S2(F) is a skyscraper.


Now let K = S1(F)/S0(G) and C = S2(F)/S1(G), so that the sequence0 → K → E → C → 0 is exact. We have deg(K) = deg(S1(F)) − deg(S0(G)) ≥ 0with equality if and only if S2(F) is a skyscraper, S0(G) is a homogeneous bundle,and deg(F) = deg(G) = 0.

Note that deg(K) = 0 since otherwise, as E is µ-semistable, one has rk(K) =rk(E), so that deg(C) ≥ 0 and deg(K) ≤ 0 which is a contradiction unless deg(K) =0. Therefore deg(F) = 0 so E is µ-semistable.

We prove now Theorem 3.35. Since H0(X, E ⊗ Pξ) = H2(X, E ⊗ Pξ) = 0for all ξ ∈ X as E is µ-stable, locally free of zero degree, E is IT1. Let again0 → F → E → G → 0 be a destabilizing sequence, this time with G µ-stable. Wemay repeat the proof of Proposition 3.38 but this time if K is a proper subsheafof E we have deg(K) < 0, so that necessarily either K = 0 or rk(K) = rk(E).

Note that S2(G) = 0, so that H2(X,G ⊗ Pξ) = 0 for all ξ ∈ X, while on theother hand H0(X,G∗∗ ⊗ Pξ) = 0 since G∗∗ is µ-stable of nonpositive degree, sothat H0(X,G ⊗ Pξ) = 0 and G is IT1. Hence we have an exact sequence

0→ S1(F)→ E → G → S2(F)→ 0 . (3.14)

If K = 0, then S1(F) = 0 and S2(F) is a skyscraper sheaf; but the sequence(3.14) implies that E is not locally free, whence we may exclude this case. Thusrk(K) = rk(E). But K ' S1(F) and since G is locally free, the morphism E → Gvanishes, and G ' S2(F) which is absurd because the first sheaf is WIT1 whilethe second is IT0.

3.5.3 Symplectic morphisms of moduli spaces

The preservation of stability expressed by Theorem 3.35 provides morphisms be-tween different components of the moduli space of stable sheaves on an Abeliansurface. These morphisms happen to be symplectic with respect to the so-calledMukai’s symplectic structure on the moduli space [225]. Let X be a (complex)Abelian or K3 surface equipped with a polarization H, and let M = MH(v) bethe moduli space of H-stable sheaves on X with Mukai vector v = (r, `, s). If thegreatest common divisor of the numbers r, ` ·H and s is 1, then M is a smoothquasi-projective variety. If [F ] is a point in M , the tangent space T[F ](M) may beidentified with the vector space Ext1(F ,F). The Yoneda product (see Eq. (A.13))provides a skew-symmetric map Ext1(F ,F) ⊗ Ext1(F ,F) → Ext2(F ,F). More-over one has a trace morphism Ext2(F ,F) → Ext2(OX ,OX) dual to the naturalmorphism C = Hom(OX ,OX) ε−→ Hom(F ,F). This defines a holomorphic 2-form ς

on M which turns out to be closed and nondegenerate, thus defining a holomorphicsymplectic 2-form.


Actually, the previous isomorphisms come from Serre duality, which involvesthe choice of an isomomorphism H2(X,ωX) ' C, which in this case is providedby integration on X of a form of type (2, 2) representing the cohomology class.The morphism Ext2(F ,F)→ C is the composition

Ext2(F ,F) trace−−−→ H2(X,OX) ' H0,2(X,C) λ−→ H2,2(X,C) ' C (3.15)

where λ is the wedging by a symplectic form ω on X. This may also be written asthe composition

Ext2(F ,F) ∼→ Hom(F ,F)∗ ∼→ C (3.16)

where the first isomorphism is Serre duality and the second is the dual of theisomorphism ε introduced before.

Remark 3.39. Let us note for future use that if X is a symplectic variety ofdimension n > 2, we can anyway define symplectic structures on the modulispaces of stable sheaves on X as above, by taking for λ in Equation (3.15) thewedging by ωm ∧ ωm−1, where ω is the symplectic form of X, and m = n/2 [184].4

Let X be an Abelian surface with a polarization H, and let MH(v) be themoduli space of H-stable sheaves on X with Mukai vector v. Let Mµ

H(v) be thesubset of MH(v) formed by locally free µ-stable sheaves on X. It is known thatMµH(v) is open in MH(v) in the Zariski topology [155]. We shall use the same

notation for the moduli space of sheaves on the dual Abelian variety X equippedwith the polarization H defined in Section 3.4.2. If ` · H = 0, by Theorem 3.35we have a morphism ς : Mµ

H(v)→MH(v)µ, induced by the Abelian Fourier-Mukaitransform S. Here, as consequence of Equation (3.2), we have v = −(s, `, r) ifv = (r, `, s) (having identified H2(X,Z) with H2(X,Z)).

Proposition 3.40. The morphism ς is symplectic.

Proof. One has a commutative diagram

Ext1(F ,F)⊗ Ext1(F ,F) //

S×S

Ext2(F ,F) //

S

Hom(F ,F)∗ ' C

Ext1(S(F),S(F))⊗ Ext1(S(F),S(F)) // Ext2(S(F),S(F)) // Hom(S(F),S(F))∗ ' C .

S∗

OO

The first square commutes by (A.14), and the second by the compatibility of theintegral functors with Serre duality.

We shall observe in Chapter 4 that a similar result holds in the case of theFourier-Mukai transform on K3 surfaces.


3.5.4 Embeddings of moduli spaces

The integral functor Φ0 : Db(C) → Db(J(C)) defined in Section 3.4.3, where Cis a smooth projective curve of genus g > 1, and J(C) its Jacobian variety, hasthe property of preserving the stability of bundles it acts on, according to a resultof Li. We may use this result to construct an embedding of the moduli space ofstable bundles on C as a subvariety of a component of the moduli space of stablebundles on J(C), which is isotropic, and in some cases Lagrangian, with respectto the natural symplectic structure of the moduli space of stable bundles on J(C).Here we basically follow [76].

We start by recalling Li’s result [196, Thm. 4.11]. For the proof we refer thereader to Li’s paper.

Proposition 3.41. Let C be a smooth projective curve of genus g > 1. If E is astable bundle on C of rank r and degree d such that d > 2rg, then E is WIT0 withrespect to the integral functor Φ0, and the transformed sheaf E = Φ0(E) is locallyfree and µ-stable with respect to the polarization given by the Θ divisor on J(C).

If E has rank r and degree d, the Chern character of E is readily computed,obtaining

ch(E) = (d+ r(1− g),−rΘ, 0, . . . , 0) (3.17)

where Θ is the cohomology class associated with the Θ divisor in J(C). Thereforeby Proposition 3.41 we obtain a map

j : MC(r, d)→MµJ(C)(r, d) , (3.18)

where MC(r, d) is the moduli space of stable sheaves on C with rank r and degreed, and Mµ

J(C)(r, d) is the subset of the moduli space of Θ-stable sheaves on J(C)with Chern character as in (3.17) that is formed by µ-stable locally free sheaves.

Proposition 3.42. Let d > 2rg. The morphism (3.18) is an embedding (i.e., both jand its tangent map are injective).

Proof. Let α : C → J(C) be the embedding given by the Abel map. This inducesa functor α : Db(C)→ Db(J(C)). Then we have an isomorphism of functors Φ0 'ΦPJ(C)→J(C) α.

On the other hand, if E is a stable bundle on C of rank r and degree d, thedirect image α∗(E) is a stable (in Simpson’s sense) pure sheaf of dimension 1 onJ(C), with Chern character

ch(α∗(E)) = (0, . . . , 0,−rΘ, d+ r(1− g)

(see Section C.2). Since ΦPJ(C)→J(C) is an equivalence of categories, the injectivityof j follows from the fact that α∗(E) ' α∗(F) implies E ' F .


As far as the injectivity of the differential of j is concerned, we note thatthe latter may be regarded as a map Ext1

C(E , E) → Ext1J(C)(α∗(E), α∗(E)). If an

extension 0 → E → F → E → 0 maps to zero, then α∗(F) ' α∗(E) ⊕ α∗(E), andthis implies F ' E ⊕ E .

Proposition 3.43. Assume that d > 2rg, where g is the genus of C. If g is even,and the map j embeds MC(r, d) into the smooth locus M0

J(C)(r, d) of MµJ(C)(r, d),

the subvarieties MC(r, d) are isotropic with respect to any of the symplectic formsdefined in Remark 3.39. In particular, when g = 2, the subvarieties MC(r, d) areLagrangian with respect to the Mukai form of Mµ

J(C)(r, d).

Proof. The symplectic form on the moduli space MµJ(C)(r, d) (cf. Section 3.5.3)

vanishes on the image of MC(r, d) because the Yoneda map

Ext1(F ,F)⊗ Ext1(F ,F)→ Ext2(F ,F)

vanishes when F is of the form F = α∗(E) for a vector bundle E on C. This isshown by a direct computation; details may be found in [247].

In the case g = 2, the moduli space is smooth by the results in [225]; moreover,

dim MµJ(C)(r, d) = 2(r2 + 1) = 2 dimMC(r, d) .

Remark 3.44. If we consider the moduli space MC(r, ξ) of stable bundles on C ofrank r and fixed determinant isomorphic to ξ, then the result is trivial: the varietyMC(r, ξ) is Fano, so that it carries no holomorphic forms. 4

Let us now briefly elaborate on the case g = 2. One can characterize situationswhere the moduli space Mµ

J(C)(r, d) is compact. This happens for instance in thefollowing case.

Proposition 3.45. Assume g = 2, d > 4r and that ρ = d − r is a prime number.Then every Gieseker-semistable sheaf on J(C) with Chern character (d−r,−rΘ, 0)is µ-stable. Moreover, if d > r2 + r, every such sheaf is locally free (this alwayshappens when r = 1, 2, 3).

Proof. Since d−r is prime, every sheaf in MJ(C)(r, d) is properly stable. Let [F ] ∈MJ(C)(r, d) and assume that the subsheaf G destabilizes F . Let ch(G) = (σ, ξ, s).Standard computations show that if F is not µ-stable, then

ξ ·Θσ

= −2rρ

and s < 0 .


Setting n = ξ · Θ we have |n| = 2rσ/ρ, with σ < ρ and ρ > 3r. This is impos-sible whenever ρ is prime. The statement about local freeness follows from theBogomolov inequality: from the exact sequence

0→ F → F∗∗ → Q→ 0

we have ch(F∗∗) = (d− r,−rΘ, λ) where λ is the total length of the torsion sheafQ. Since F∗∗ is µ-stable it satisfies the Bogomolov inequality [155], which in thiscase reads r2 ≥ λ(d− r). Together with the condition d > r2 + r this forces λ = 0,i.e., F is locally free.

Remark 3.46. In the case g = 2, the complex Lagrangian embedding j : MC(r, d)→MµJ(C)(r, d) provides examples of special Lagrangian submanifolds. We refer the

reader to [76] for this aspect. 4Remark 3.47. In the case g = 4, one can use the fact that MC(r, d) embedsisotropically into Mµ

J(C)(r, d) to show that at a smooth point [E ] of MµJ(C)(r, d)

one has h2(End0(E)) > 0, that is, the sufficient condition for the smoothness of thespace Mµ

J(C)(r, d) is not satisfied. Let us set A(r) = dimh2(End0(E)) (as we shallsee this depends only on the rank). The isotropicity condition yields the inequality

dimMJ(C)(r, d) ≥ 6r2 + 2.

On the other hand, if [E ] is a smooth point of the moduli space corresponding toa vector bundle E , we have

dimMJ(C)(r, d) = h1(End(E))

= 1 + 3 + 12A(r) ≥ 6r2 + 2,

whence the inequalityA(r) ≥ 12r2 − 4 > 0

follows. 4


A fairly comprehensive treatment of the Abelian Fourier-Mukai transform is givenby A. Polishchuk in the book [251], which also contains an extensive bibliography.

It is also worth mentioning that the classical Torelli theorem about the char-acterization of a projective curve in terms of its Jacobian has been proved by usingthe Abelian Fourier-Mukai transform in [36]. A generalization of that paper to thecharacterization of curves from their Prym varieties is contained in [233].


The Abelian Fourier-Mukai transform was used by Beauville [32] to studythe Chow ring of an Abelian variety.

The group of autoequivalences of the derived category of an Abelian varietyX over a field k has been characterized by Orlov in [243] as an extension of thegroup of isometric isomorphisms of X ×X by Z⊕ (X × X)k, where (X × X)k isthe group of rational points of X × X.

The preservation of Gieseker stability under the Abelian Fourier-Mukai trans-form has been studied by Maciocia in the case of Abelian surfaces [203]. The resultsare largely negative: Gieseker stability is not well-behaved under the action of thetransform. The situation improves if one considers a notion of twisted Giesekerstability, as shown by Yoshioka [294, 295].

Chapter 4

Fourier-Mukai on K3 surfaces

Introduction

In looking for examples of Fourier-Mukai transforms on varieties other than theAbelian ones, it is natural to consider K3 surfaces, especially in view of Theorem2.38 and the subsequent discussion.

A forerunner of a Fourier-Mukai functor for K3 surfaces (which in our no-tation is a morphism of the type fQ : H•(X,Z) → H•(Y,Z), cf. Eq. (1.12)) wasintroduced by Mukai in [227]. When trying to define a Fourier-Mukai functor inthe proper sense, one realizes that it is necessary to limit the class of K3 surfacesone considers; essentially one needs to require that the Picard lattice contains somepreferred sublattice. A first example was given in [24] where a class of K3 surfacescalled (strongly) reflexive was introduced. Another example by Mukai appearedlater [228].

In this chapter we start by giving a general introduction to K3 surfaces,considering at first the general Kahlerian case and then specializing to the alge-braic case. We also provide the basic elements of the theory of moduli spaces ofstable sheaves on K3 surfaces. The core part of the chapter is the constructionof the Fourier-Mukai transform for reflexive K3 surfaces. The transformation isconstructed by realizing a reflexive K3 surface X as a fine moduli space of sta-ble bundles on X itself, and using the corresponding universal bundle as integralkernel. Applications are then given to the study of a class of bundles, called homo-geneous, which extend the notion of homogeneous bundles on Abelian varieties,

contains Mukai’s example, which is very similar in nature.

Section 4.4 shows that the Fourier-Mukai transform for reflexive K3 sur-


and to the Hilbert schemes of points of a (reflexive) K3 surface. Section 4.3.4 also


112 Chapter 4. Fourier-Mukai on K3 surfaces

faces shares another important feature with the Abelian Fourier-Mukai transform,namely, under suitable conditions it preserves the stability of the sheaves it actson.

The base field is always C and all characteristic classes take values in the coho-mology ring. For simplicity, in this chapter by “(semi)stable” we mean “Gieseker-(semi)stable.”

4.1 K3 surfaces

Here we wish to collect some basic results on K3 surfaces in the most generalsetting, i.e., we do not assume they are algebraic (which is not always the case,of course) or Kahlerian (which on the other hand is always the case by a well-known theorem of Y.-T. Siu [270], cf. our Theorem 4.7). We denote by TX the(holomorphic) tangent bundle of X and ΩpX the sheaf of germs of holomorphicp-forms, i.e., the sheaf of sections in the locally free sheaf ΛpT ∗X .

Definition 4.1. A K3 surface is a compact connected smooth complex surface Xsuch that q = dimH1(X,OX) = 0 and the canonical bundle ωX is trivial. 4

It follows from the definition that pg = dimH0(X,ωX) = 1 and c1(X) =−c1(ωX) = 0. By using Noether’s formula

1− q + pg =112

(c1(X)2 + c2(X))

we get c2(X) = 24 (as usual, we have identified H4(X,Z) with Z by integrat-ing over the fundamental class of X). As a consequence, one obtains that thetopological Euler characteristic χtop(X) =

∑i(−1)ibi is equal to 24. Since b1 =

dimH1(X,R) is either 2q or 2q − 1 according to whether it is even or odd [22,Theorem IV.2.7], one necessarily has b1 = 0.

Proposition 4.2. One has H1(X,Z) = H1(X,Z) = H3(X,Z) = H3(X,Z) = 0, andH2(X,Z) ' H2(X,Z) is a free Z-module of rank 22.

Proof. The only nontrivial fact to prove is that H1(X,Z) = 0. Since we alreadyknow that H1(X,R) = 0, then H1(X,Z) may only be a torsion module. But theexistence of a torsion element of order k > 1 implies the existence of a compactcomplex surface Y which is a k-fold covering of X and would violate Noether’sformula. By the universal coefficient theorem, the torsion submodule of H2(X,Z)is isomorphic to the torsion submodule of H1(X,Z).

4.1. K3 surfaces 113

The intersection form

H2(X,Z)×H2(X,Z)→ Z(γ, γ′) 7→ γ · γ′

defines a lattice structure on H2(X,Z). Poincare duality is precisely the statementthat this lattice is unimodular. The index τ(X) = b+ − b− of the intersection isreadily computed through the Hirzebruch formula:

τ(X) =13

(c21 − 2c2) = −16 .

So, we get b+ = 3 and b− = 19.

Since the second Stiefel-Whitney class of X vanishes (indeed, w2(X) =c1(X) mod 2 = 0), Wu’s formula

γ · γ = w2 · γ mod 2 for all γ ∈ H2(X,Z)

tells us that the intersection form is even. The classification theorem B.2 for in-definite unimodular even lattices yields the following result.

Proposition 4.3. The Z-module H2(X,Z) of a K3 surface X endowed with theintersection form is a lattice isomorphic to Σ = U ⊕ U ⊕ U ⊕ E8〈−1〉 ⊕ E8〈−1〉,where U is the rank 2 hyperbolic lattice, and E8 is the rank 8 lattice whose in-tersection form is the Cartan matrix associated to the exceptional Lie algebra e8

(cf. Eq. (B.1)).

Example 4.4. Let us consider a smooth quartic surface X in P3. By Bott’s homo-topic version of Lefschetz’s hyperplane theorem [56], the fundamental group of Xis trivial. Hence, 0 = b1(X) = 2q(X). Moreover, a direct computation using theadjunction formula [131, p. 146] gives c1(X) = 0. But, since q(X) = 0, the firstChern class classifies holomorphic line bundles on X. Thus, the canonical bundleis trivial, and X is a K3 surface. 4

Example 4.5. An important class of K3 surfaces is provided by the Kummer sur-faces. Let us consider the involution ι : (z, z′) 7→ (−z,−z′) on a complex 2-torusT = C2/Λ. The quotient X ′ = T/ι has 16 conical singularities corresponding tothe fixed points of the involution. Blowing up these singularities one gets a smoothsurface X, having b1(X) = 0 and trivial canonical bundle (see [22, V.16]). 4

Example 4.6. Let C be a smooth sextic in P2; one has g(C) = 10. To the sexticC we associate a section τ ∈ H0(P2,OP2(6H)) such that (τ) = C; we fix anisomorphism OP2(3H) ⊗ OP2(3H) ' OP2(6H). In the total space of the fibrationp : OP2(3H) → P2 we consider the locus X = (x, λ)|λ ⊗ λ = τ(x). So, Xis a smooth double cover of P2 branched along C. By the Hurwitz formula the


canonical bundle of X is trivial. Moreover, we have p∗OX ' OP2 ⊕OP2(−3H) [22,I.17.2]. Since p is a finite morphism, the sheaves OX and p∗OX have isomorphiccohomologies, so that H1(X,OX) = 0, and X is a K3 surface. 4

According to a general result, sometimes called the Kodaira conjecture, acompact complex surface admits a Kahler metric if and only if b1(X) is even[77, 192] (see also [22, IV.3.1]). Since for a K3 surface b1 = 0, one has at once thefollowing result.

Theorem 4.7. Every K3 surface X admits a Kahler metric.

This result was first proved by Siu in 1983 [270]. It implies that the decom-position of the cohomology space H2(X,C) of a K3 surface X

H2(X,C) = H2,0(X)⊕H1,1(X)⊕H0,2(X) (4.1)

(which can be defined on any compact complex surface for the Frohlicher spec-tral sequence degenerates at E1-level [22, IV.2.8]) coincides with the usual Hodgedecomposition for any choice of the Kahler metric. The Hodge numbers hp,q =dimC H

p,q(X) are readily calculated. One has h0,2 = h2,0 = 1, and h1,1 = b2 −h2,0 − h0,2 = 20. The C-linear extension of the intersection form on H2(X,C)coincides with the cup product of cohomology classes of differential forms. Itsrestriction to H1,1(X) ∩H2(X,R) has signature (1, h1,1 − 1) = (1, 19).

For any algebraic surface, one denotes by Pic(X) = H1(X,O∗X) the Picardgroup of isomorphism classes of line bundles over X. By the Lefschetz theoremon (1, 1) classes, the image of Pic(X) in H2(X,Z) coincides with the intersectionH1,1(X)∩j∗(H2(X,Z)), where j∗ : H2(X,Z)→ H2(X,C) is the natural injection.This sublattice of H2(X,Z) is called the Neron-Severi group of the surface X. Itsrank is called the Picard number of X. If X is a K3 surface one has q = 0, hencethe map c1 : Pic(X)→ H2(X,Z) is injective, and one can identify Pic(X) with theNeron-Severi group. We shall often use the common terminology Picard lattice.

A useful characterization of Pic(X) is provided by the following criterion,which is an easy consequence of the orthogonality of the Hodge decomposition(4.1).

Proposition 4.8. A class in H2(X,Z) is in Pic(X) if and only if it is orthogonalto H2,0(X).

Definition 4.9. The orthogonal complement to Pic(X) in H2(X,Z) is called thetranscendental lattice, which will be denoted by T(X). 4

A class d ∈ Pic(X) is said to be nodal if d2 = −2. The Riemann-Rochtheorem implies that d or −d is the class of a unique curve C which is irreducible[22, Prop. VIII.3.7]. The curve C is said to be nodal and is smooth and rational.

4.1. K3 surfaces 115

The importance of nodal curves in the theory of K3 surfaces stems from thefact that the set of effective classes on a K3 surface is the semigroup generated bythe nodal classes and the integral points in the closure of the positive cone [22,Prop. VIII.3.8].

The lattice H2(X,Z) endowed with its natural weight-two Hodge structuregiven by the decomposition 4.1 contains essentially all the information about thegeometric structure of the K3 surface X. The choice of an isometry φ : H2(X,Z)→Σ is called a marking of the K3 surface X. The line φ(H2,0(X,C)) ⊂ Σ ⊗ Cdetermines a point (called the period of X) in the period domain ∆ ⊂ P(Σ⊗ C),which is defined by the equations

α · α = 0 , α · α > 0

(this assignment is called the period map, and is surjective, cf. [22]). Here α is theimage under φ of a generator of H2,0(X,C)). According to the idea underlying theTorelli theorem, two K3 surfaces X and Y are isomorphic if and only if they can begiven markings such that the corresponding points in period domain coincide. Ac-tually, this statement can be strengthened to a more precise and deeper result, the(global) Torelli theorem, which was first proved by Pjateckiı-Sapiro and Safarevic[248] in the projective case and by Burns and Rapoport [81] in the Kahler case(see [91] or [22] for a detailed account).

Let X and Y be K3 surfaces. A group isomorphism φ : H2(X,Z)→ H2(Y,Z)is a Hodge isometry if it preserves both the intersection forms and the naturalHodge structures (the second requirement is equivalent to saying that the C-linear extension of φ to H2(X,C) preserves the Hodge decomposition). A Hodgeisometry is said to be effective if it maps the Kahler cone of X to the Kahler coneof Y . By means of the notion of Hodge isometry, one can state a Torelli theoremfor K3 surfaces, both in strong and weak form [22, Thm. 11.1, Cor. 11.2].

Theorem 4.10. (Torelli theorem) Let X and Y be K3 surfaces, and let φ : H2(Y,Z)→ H2(X,Z) be an effective Hodge isometry. Then, there is a unique isomorphismf : X → Y such that f∗ = φ.

A weaker form of this result, which follows from the previous one, will beuseful in the sequel.

Corollary 4.11. (Weak Torelli theorem) Let X and Y be K3 surfaces whose latticesH2(X,Z) and H2(Y,Z) are Hodge isometric. Then X and Y are isomorphic.

The Torelli theorem is a consequence of the fact that there is a well-behavedtheory of deformations for K3 surfaces, both local and global. In particular, itturns out all K3 surfaces are deformation equivalent to each other, so that theyare all diffeomorphic as differentiable 4-manifolds. This shows that every K3 is


simply connected, since this is the case for the particular example of the quarticsurface in P3 (Example 4.4).

By using the Torelli theorem one is able to construct a universal familyof (Kahlerian) marked K3 surfaces. Its base space (the moduli space of markedKahlerian K3 surfaces) is a smooth, non-Hausdorff analytic space of dimension 20[22]. Algebraic marked K3 surfaces form a subfamily of dimension 19.

4.2 Moduli spaces of sheaves and integral functors

In this section we introduce moduli spaces of sheaves on algebraic K3 surfaces;when such moduli space are fine, so that they carry a universal sheaf, the lattermay be used as the kernel of a Fourier-Mukai transform.

We need to fix some preliminary notation and definitions.

The Todd class of a K3 surface X is

td(X) =(1,

12c21(X),

112

(c21 + c2(X)))

= (1, 0, 2) ∈ H2•(X,Z) ,

so that√

td(X) = (1, 0, 1). Hence, the Mukai vector of a coherent sheaf E onX is the element v(E) = ch(E)

√td(X) = (rk E , c1(E), χ(E) − rk(E)), which still

belongs to the even part of the integer cohomology of X. We recall that the Mukaipairing (Definition 1.3) is a symmetric bilinear form 〈·, ·〉 on the even part of thecohomology of X that for a K3 surface takes the form

〈v, w〉 = −∫X

v∗ · w ,

where v∗ = (v0,−v1, v2) is the Mukai dual of v.

Definition 4.12. Let

H0,2(X) = H0,2(X) , H2,0(X) = H2,0(X)

H1,1(X) = H0(X)⊕H1,1(X)⊕H4(X) .

Then the resulting weight-two structure H2•(X,Z), Hp,q(X) is denoted byH•(X,Z). 4

The natural inclusion H2(X,Z) → H•(X,Z) respects the Hodge structures.The space H•(X,Z) endowed with the Mukai pairing 〈·, ·〉 is an even lattice iso-morphic to Σ⊕U (so it has signature (4, 20)). The restriction of the Mukai pairingto H2(X,Z) coincides with the intersection product. We shall also use the notation

H1,1(X,Z) = H1,1(X) ∩H2(X,Z) ' Pic(X)⊕ U .

4.2. Moduli spaces of sheaves and integral functors 117

Since the canonical sheaf of X is trivial, Serre duality for Ext groups togetherwith Equation (1.6) yields the formula

〈v(E), v(E)〉 = dim Ext1(E , E)− 2 dim HomOX (E , E) (4.2)

for any coherent sheaf E on X. From this we see that dim Ext1(E , E) is alwaysan even integer, for the Mukai pairing is even. If the sheaf E is simple, thenHomOX (E , E) ' C, so that v2(E) ≥ −2. The space Ext1(E , E) is canonically iso-morphic to the Zariski tangent space to the infinitesimal deformations of E .

We fix a polarization H on X and we consider the (coarse) moduli spaceMH(v)ss of S-equivalence classes of sheaves which are semistable with respect toH and have Mukai vector equal to v. By Maruyama’s general results [211, 212],we know that MH(v)ss is a (possibly empty) projective variety. In most applica-tions and examples we will be more interested in studying an open subscheme ofMH(v)ss, namely the (coarse) moduli space MH(v) parameterizing stable sheaves.

The following fundamental fact follows from results due to Mukai [225].

Theorem 4.13. Let v be an element in H1,1(X,Z). The moduli space MH(v) ofstable sheaves whose Mukai vector is v is a smooth scheme of dimension v2 + 2(possibly empty). The canonical bundle of MH(v) is trivial.

Corollary 4.14. If v = (r, `, s) with v2 = 0 and r ≥ 2, and E is a µ-stable sheaf onX (with respect to some polarization) with v(E) = v, then E is locally free.

Proof. If E is not locally free, its double dual E∗∗ has Mukai vector v′ = (r, `, s+λ),where λ is the length of the support of the quotient E∗∗/E . Since E∗∗ is µ-stable,the moduli space MH(v) is nonempty of dimension (v′)2 + 2 = −2rλ + 2. Sincethis cannot be negative, one must have λ = 0, i.e., E is locally free.

Mukai’s results provide a quite exhaustive description of the moduli spacesof dimension 0 or 2. We need to state a technical result [227, Prop. 2.14].

Lemma 4.15. Let E be a torsion-free coherent sheaf on X. The following inequalityholds:

dim Ext1(E∗∗, E∗∗) + 2 length(E∗∗/E) ≤ dim Ext1(E , E) .

Recall that on a smooth surface the dual of a coherent sheaf is always locallyfree.

Theorem 4.16. Let v be an element in H1,1(X,Z) such that v2 = −2. If thereexists at least one stable sheaf E on X whose Mukai vector is equal to v, then themoduli space Mss

H (v) is a single reduced point and E is locally free.


Proof. Assume that F is a semistable sheaf with v(F) = v. By Equation (1.6) wehave χ(E ,F) = −〈v(F), v(E)〉 = −v2 = 2. It follows (using Serre duality) thatdim Hom(E ,F) + dim Hom(F , E) > 0. Since E stable and F are semistable of thesame Hilbert polynomial, we conclude that E ' F . Now Lemma 4.15 implies thatlength(E∗∗/E) = 0, so that E is locally free.

Remark 4.17. Kuleshov shows in [191] that for every Mukai vector v = (r, `, s),with r > 1 and v2 = −2, there exists a simple µ-semistable bundle E such thatv(E) = v. 4

We now address the issue which has the greatest interest to us, namely thecase of the two-dimensional components of the moduli space. The results provedby Mukai in the papers [225, 227] suggest the existence of a notion of “duality”for K3 surfaces similar, under many respects, to that holding for Abelian varieties.This idea has been further developed in [24, 26, 228].

In general, the moduli spaces MH(v) and MssH (v) are not irreducible, and the

first is strictly contained into the latter. However in dimension 2 the existence ofa compact component implies that every semistable sheaf is actually stable andthat the moduli space is irreducible. The following result is a consequence of [227,Prop. 4.4] and of Theorem 4.13 (a concise proof can be found in [155, p. 144]).

Theorem 4.18. Let v ∈ H1,1(X,Z) such that v2 = 0. Suppose there exists a com-ponent M of the moduli space MH(v) which is compact and irreducible. ThenM = MH(v) = Mss

H (v), and this surface is smooth, irreducible and compact withtrivial canonical bundle. Therefore, it is either an Abelian surface or a K3 surface.

A universal family for the moduli of stable sheaves on a projective variety maynot exist (the definition of universal family is given below). In order to circumventthis obstacle, one introduces the notion of quasi-universal family (see [227]). LetS be a scheme and let M be a connected component of the (coarse) moduli spaceof stable sheaves on S (with respect to a fixed polarization). We denote by y apoint in M and by Ey the corresponding stable sheaf on S.

Definition 4.19. A coherent sheaf Q on S ×M is a quasi-universal family if thefollowing conditions are satisfied:

1. Q is flat over M ;

2. for all y ∈M there exists an positive integer ν such that Qy ' E⊕νy ;

3. for every scheme T and for every sheaf Q′ over S × T flat over T withQ′t ' Eν

′

t for some stable sheaf Et ∈M and for all t ∈ T , ν′ being a positiveinteger independent of t, there exists a unique morphism u : T →M and twolocally free sheaves F and F ′ on T such such that u∗Q⊗F ' Q′ ⊗F ′.

4


The number ν is called the similitude of the family. A quasi-universal familywith ν = 1 is nothing but a universal family in the usual sense. Building onwork by Maruyama [212], Mukai proved a general result on the existence of aquasi-universal family for the moduli functor of simple sheaves [227, Thms. A.5,A.6], which we state only in the case of K3 surfaces, and for the moduli of stablesheaves.

Theorem 4.20. Let M be a connected component of the moduli space MH(v) ofstable sheaves over a K3 surface X. Then there exists a quasi-universal family Qon X×M . Moreover, if the greatest common divisor of the integers rk(E), c1(E)·Dand χ(E) (where E ∈M and D runs over all divisors in X) is 1, then there existsa universal family on X ×M and M is a fine moduli space.

We may now use a universal family Q as a kernel to define an integral functor

ΦQM→X : Db(M)→ Db(X) .

Proposition 4.21. Assume that dimM = 2. The integral functor is fully faithful ifand only if the family Q is universal. Moreover, in this case both functors

ΦQM→X : Db(M)→ Db(X) and ΦQX→M : Db(X)→ Db(M)

are Fourier-Mukai transforms.

Proof. If Q is universal, then for every y ∈ M the sheaf Qy is simple. Moreover,if y1, y2 are distinct points in M , we have Hom(Qy1 ,Qy2) = Ext2(Qy1 ,Qy2) = 0,while Ext1(Qy1 ,Qy2) = 0 since v2 = 0. So Q is strongly simple over M , and thefunctor ΦQM→X is fully faithful by Theorem 1.33.

Conversely, if ΦQM→X is fully faithful, then Q is strongly simple over M , sothat all sheaves Qy are simple and the similitude ν must be 1.

The functors ΦQM→X and ΦQX→M are exact equivalences by Corollary 2.58.

Remark 4.22. If Q is a quasi-universal it may be convenient to regard the modulispace as a stack (or gerbe). In this picture, it becomes a fine moduli space. See,e.g., [82, 83, 37]. For a “gerby” Fourier-Mukai transform see [97]. See also Section6.7. 4

If Q is universal and locally free, we have Φ Φ ' [−2] and Φ Φ ' [−2],where Φ = ΦQX→M and Φ = ΦQ

∗

M→X . In this case one has the results expressed by thefollowing corollaries.

Corollary 4.23. If F is WITi with respect to Φ, then F is WIT2−i with respect to

Φ. Moreover F ' F .


The spectral sequence (2.35) takes the form

Ep,q2 = Φp(Φq(E)) =⇒

E if p+ q = 20 otherwise .

(4.3)

Corollary 4.24. For every sheaf F on X, the sheaf Φ0(F) is WIT2 with respect toΦ, while the sheaf Φ2(F) is WIT0 with respect to Φ (and hence IT0 by Proposition1.7).

We can use the integral functor ΦQX→M , when the family Q is universal, as ahandy tool to study the two-dimensional components of the moduli space of stablesheaves on X. When v2 = 0, Theorem 4.18 implies that the existence of a compactand irreducible component of MH(v) is equivalent to MH(v) itself being compactand irreducible.

The following result was originally proved by Mukai [227]. Our proof relieson techniques developed in Chapter 1.

Theorem 4.25. Assume v2 = 0. If MH(v) is compact and there exists a universalfamily Q on X ×MH(v), then MH(v) is a K3 surface.

Proof. Let us write M for MH(v). By Proposition 4.21 the functor ΦQX→M is anequivalence of triangulated categories. Hence, by Corollary 2.40 the induced mapin cohomology

f = fQ : H•(X,Q)→ H•(M,Q)

α 7→ πM∗(π∗Xα · v(Q))

is an isomorphism of Q-vector spaces. By Theorem 4.18, M is either an Abelian ora K3 surface, but the first case cannot occur since X is K3 and the cohomologiesof the two kinds of surfaces are different.

Note that in view of Equation (1.13), the inverse map f−1 is f−1 = fQ∨

(weneed to take the dual in the derived category because Q may fail to be locallyfree).

In the hypotheses of Theorem 4.25 we denote by X the moduli space MH(v).The functor Φ = ΦQ

X→X is a Fourier-Mukai transform by Proposition 4.21 andTheorem 4.25.

We now prove that the map f , defined in the proof of Theorem 4.25, inducesan isometry between the lattices H•(X,Z) and H•(X,Z).

Lemma 4.26. [227, Lemma 4.7] The Mukai vector v(F) is integral for any sheafF on X × X. As a consequence, the Mukai vector v(E•) is integral for any objectE• ∈ Db(X × X).


Proof. This amounts to proving that ch(F) is integral. Let us denote by ψi,j the(i, j) Kunneth component of ψ = ch(F). As X and X have trivial canonicalbundles, (ψ2,0)2 and (ψ0,2)2 are even, so that ch2(F) is integral. Now one has

(ψ · π∗X

td(X))2i,4 =∑j

(−1)i chi(RjπX∗F)⊗ $ ,

where $ is the fundamental class of X. This implies that ch4(F) and ψ2,4 areintegral. Interchanging the roles of X and X one shows that ψ4,2 is integral. Thesecond statement is straightforward since X is smooth.

Proposition 4.27. [227, Thm. 1.5][242, Prop. 3.5]

1. The map f yields a Hodge isometry f : H•(X,Z) ∼→ H•(X,Z).

2. f(v∗) = $, where $ is the fundamental class of X.

3. f induces an isometry (v⊥/v, 〈 , 〉) ∼→ H2(X,Z).

Proof. 1. By Lemma 4.26, the Mukai vector v(Q) is integral and then the map f

is defined over Z. Using the projection formula, one has

〈w, f(u)〉 = −∫X

w∗f(u) = −∫X×X

π∗X

(w∗)v(Q)π∗X(u)

= −∫X×X

π∗X

(w)v(Q)∗π∗X(u∗) = −∫X

f−1(w)u = 〈f−1(w), u〉 ,

as v(Q)∗ = v(Q∨). This shows that f is a Hodge isometry.

2. For a fixed y ∈ X, we have ΦQ∨

X→X(Oy) = Lj∗y(Q∨) ' (Qy)∨, where Qy =Q|X×y. This in turn implies that ΦQ

X→X((Qy)∨) ' Oy. By the commutativity ofthe diagram (1.11) one has f(v∗) = v(Oy) = $.

3. Since the Mukai dual v 7→ v∗ is an isometry, v⊥/Zv is isometric to(v∗)⊥/Zv∗. By the previous point, there are isometries

(v∗)⊥/Zv∗ ∼→ $⊥/Z$ ∼→ H2(X,Z) .

Corollary 4.28. [227, p. 347] The map f yields a Hodge isometry between thetranscendental lattices of X and X, i.e., f|T(X) : T(X) ∼→ T(X).

Proof. Since v(Q) is integral, the map f provides Hodge isometries H1,1(X,Z) ∼→H1,1(X,Z) and T(X) ∼→ T(X).


Orlov has proved that the existence of such an isometry between the trascen-dental lattices of two K3 surfaces X and Y is equivalent to the fact that X and Yhave equivalent derived categories. We shall prove this result in Chapter 7 (The-orem 7.24).

4.3 Examples of transforms

In this section we consider the case where the “partner” of the K3 surface is a2-dimensional moduli space of µ-stable locally free sheaves and the integral kernelis provided by a universal family on the product. Other Fourier-Mukai transformsfor K3 surfaces will be introduced later on. We start with some technical results.

Let X be an algebraic K3 surface with a polarization H. We have the fol-lowing existence result (cf. [227, Thms. 5.1, 5.2]). Recall that a Mukai vector v isprimitive if it is not an integer multiple of any other Mukai vector.

Lemma 4.29. If v = (r, `, s) is a primitive Mukai vector with v2 = 0 and r ≥ 1,there exists a µ-semistable simple sheaf E on X with v(E) = v. Moreover, for everydivisor class D of X, E can be chosen so that

D · c1(F)rk(F)

≥ D · `rk(E)

for every torsion-free rank 1 quotient sheaf F of E such that µ(E) = µ(F).

We reproduce from [26] the following lemma.

Lemma 4.30. Let E be a simple µ-semistable sheaf with v(E) = (2, `, s), ` ·H = 0,v(E)2 = 0 and s odd. Then E is locally free.

Proof. Note that v(E) is primitive. If E is not locally free, [227, Prop. 3.9] impliesthat E∗∗ is rigid, i.e., Ext1(E∗∗, E∗∗) = 0, and that there is an exact sequence

0→ E → E∗∗ → Ox → 0 (4.4)

for a point x ∈ X. It follows that ch2(E∗∗) = ch2(E)+1, so that v(E∗∗) = (2, `, s+1)and v(E∗∗)2 = −4. By [227, Prop. 3.2] E∗∗ is not simple, hence is not stable. Wethen have a destabilizing sequence

0→ OX(D1)→ E∗∗ → IZ(D2)→ 0 (4.5)

where IZ is the ideal sheaf of a zero-dimensional subscheme, and D1, D2 aredivisors of degree zero such that ` = D1 +D2 and D2

1 ≥ s− 1.

4.3. Examples of transforms 123

The divisors D1, D2 are not numerically equivalent, since we would have` ≡ 2D1 and then D2

1 = s, which is absurd because s is odd; thus by the Hodgeindex theorem [22, 141],

0 > (D1 −D2)2 = D21 +D2

2 − 2D1 ·D2 = `2 − 4D1 ·D2 = 4s− 4D1 ·D2 ,

so that D1 ·D2 > s. Since c2(E∗∗) = s+ 1, the exact sequence (4.5) yields s+ 1 =D1 ·D2 + length(Z) > s+ length(Z) so that length(Z) = 0 and D1 ·D2 = s+ 1.Then, (D1 −D2)2 = −4 and we have

dim Ext1(O(D2),OX(D1)) = −χ(X,OX(D1 −D2)) = 0 ,

which implies E∗∗ ∼→ OX(D1) ⊕ OX(D2). Then, the exact sequence (4.4) impliesthat either E ∼→ Ix(D1)⊕OX(D2) or E ∼→ OX(D1)⊕Ix(D2), which is absurd sinceE is simple.

Let us now take a Mukai vector v = (r, `, s) in H1,1(X,Z), such that v2 = 0and r > 0. Moreover we assume that the moduli space X = MH(v) of stablesheaves with Mukai vector v is compact and irreducible and that there is a universalfamily Q on X × X. So X is an algebraic K3. We show that X carries a naturalpolarization.

The composition of the injection Pic(X) → H1,1(X,Z), the Hodge isometryf : H•(X,Z)∼→ H•(X,Z) and the projection ofH1,1(X,Z) onto its direct summandPic(X) defines a morphism µ : Pic(X)→ Pic(X). This may be explicitly computedas

µ(α) = πX∗(γ2,2α) (4.6)

where γi,j is the (i, j) Kunneth component of γ = ch(Q).

Proposition 4.31. Assuming that all points in X correspond to locally free sheaves,the class H = −µ(H) ∈ Pic(X) is ample.

Proof. We prove that H is ample by comparing it with the first Chern class of thedeterminant bundle

Lm = det(Φ(OX(mH)))−1 ⊗ det(Φ(OX(−mH))) ,

where Φ = ΦQX→X , which is known to be ample for m 0 by a theorem of

Donaldson [101, §5]. Indeed a simple computation using Grothendieck-Riemann-Roch shows that c1(Lm) = mH.

One can also give a transcendental proof of this fact by identifying H with apositive multiple of the class of the Weil-Petersson metric on X [24, Prop. 6].


4.3.1 Reflexive K3 surfaces

The case of the so-called reflexive K3 surfaces provides an example of an explicitconstruction of a Fourier-Mukai transform for K3 surfaces.

Definition 4.32. A K3 surface is reflexive if it carries a polarization H and a divisor` such that H2 = 2, H · ` = 0, `2 = −12, and `+ 2H is not effective. 4

Reflexive K3 surfaces are particular instances of the K3 surfaces mentionedin Example 4.6.

Our first step is to prove that for any reflexive K3 surface X there is acomponent X of the moduli space of stable sheaves on X which is a K3 surfacemade of locally free µ-stable sheaves. Let us fix the Mukai vector v = (2, `,−3).

Proposition 4.33. There exists a µ-semistable simple locally free sheaf E on X withv(E) = v such that ` · c1(F) ≥ −6 for every torsion-free rank 1 quotient sheaf Fof E of degree 0.

Proof. By Lemma 4.29, taking D = `, there is µ-semistable simple sheaf E on X

fulfilling the remaining conditions. Moreover, E is locally free by Lemma 4.30.

Proposition 4.34. There exists a stable locally free sheaf E on X with v(E) = v

(so that the moduli space MH(X, v) is not empty). Moreover, every element inMH(X, v) is locally free.

Proof. Let E be the sheaf provided by Proposition 4.33. If it is not stable, thereexists an exact sequence

0→ OX(D1)→ E → IZ(D2)→ 0 ,

where IZ is the ideal of a zero-dimensional subscheme, D1, D2 are divisors ofdegree 0, and D2

1 ≥ −5, so that D21 ≥ −4 since D2

1 is even. From ` = D1 + D2,−1 = c2(E) = D1 · D2 + length(Z) and ` · D2 ≥ −6, we obtain −4 ≤ D2

1 ≤−5 + length(Z), so that length(Z) ≥ 1. Moreover D1, D2 are not numericallyequivalent, since we would have ` ≡ 2D1 and then D2

1 = −3, which is absurd; thusby the Hodge index theorem,

0 > (D1 −D2)2 = −12− 4D1 ·D2 ,

so that D1 ·D2 > −3 and length(Z) < 2. Then length(Z) = 1 and D21 = D2

2 = −4.Since ` + 2H = (D1 + H) + (D2 + H) and D1 + H, and D2 + H are linearlyequivalent to nodal curves of degree 2, this contradicts the fact that `+ 2H is noteffective.

So the moduli space MH(X, v) is not empty. Moreover every element inMH(X, v) is locally free by Lemma 4.30.


Since the greatest common divisor mentioned in Theorem 4.20 is 1, we canapply Proposition 4.21 and Theorem 4.25 to get the following result.

Proposition 4.35. If X is a reflexive K3 surface with polarization H, and v =(2, `,−3), then there is a locally free rank 2 universal sheaf Q on X ×MH(v) →MH(v) making MH(v) a fine moduli scheme parameterizing locally free stablesheaves with vector v. Moreover, X = MH(v) is a projective K3 surface and theintegral functor ΦQ

X→X is a Fourier-Mukai transform.

Note that since

−f(H) = H + terms in H4(X,Z)

one hasH2 = f(H)2 = H2 = 2 (4.7)

because f is an isometry.

4.3.2 Duality for reflexive K3 surfaces

We have chosen to call “reflexive” the K3 surfaces previously introduced becausethey are self-dual in the sense that the moduli space X = MH(2, `,−3) is a reflexiveK3 surface as well. One can realize X as a moduli space of sheaves on X stablewith respect to the natural polarization H; the universal family on X×X, suitablynormalized, will turn out to be isomorphic toQ∗. To this end we need to strengthena little bit our assumptions about the K3 surface X.

Since for every y ∈ X the sheaf Qy = Q|X×y is a µ-stable locally free sheafof degree zero, one has h0(X,Qy) = 0, h2(X,Qy) = 0 and h1(X,Qy) = 1. So thestructure sheaf OX is IT1 and the sheaf R1πX∗Q is a line bundle on X. We cantherefore normalize the universal bundle Q by setting

R1πX∗Q = OX . (4.8)

We shall henceforth assume that this normalization has been fixed.

Definition 4.36. A K3 surface is strongly reflexive if it carries a polarization H

and a divisor ` such that H2 = 2, H · ` = 0, `2 = −12, and there are in X nonodal curves of degree 1 or 2. 4

Strong reflexivity is a generic condition. Indeed, the ample divisor H definesa double cover of P2 branched over a sextic; the image of a nodal curve of degree 1is a line tritangent to the sextic, while the image of a nodal curve of degree 2 is aconic, tangent to the sextic at six points. Neither situation can arise in the generalcase. A coarse moduli space parameterizing strongly reflexive K3 surfaces, which


is an irreducible quasi-projective scheme of dimension 18, is constructed in [25].On a generic Kummer surface one can choose divisors H and ` making it stronglyreflexive.

Strongly reflexive K3 surfaces are reflexive.

Lemma 4.37. If X is a strongly reflexive K3 surface the class E = ` + 2H is noteffective.

Proof. Since E2 = −4, if E is effective it is not irreducible and E = D + F forsome nodal curve D. Then D ·H = 3 and F ·H = 1, so that F is also irreducible.It follows that F 2 ≥ −2. If F 2 ≥ 0, then D · F ≤ −1, so that D = F which isabsurd. Thus, F 2 = −2 and F is a nodal curve of degree 1, a situation we areexcluding.

On strongly reflexive K3 surfaces the sheaves in the moduli space X = MH(v)is Proposition 4.35 are µ-stable.

Proposition 4.38. Any stable bundle E on X with v(E) = (2, `,−3) is µ-stable.

Proof. If E is not µ-stable, it can be destabilized by a sequence

0→ OX(D1)→ E → IZ(D2)→ 0 ,

where IZ is the ideal of a zero-dimensional subscheme, D1, D2 are divisors ofdegree 0, and D1 ·D2 = −1− length(Z). Since E is stable,

D21 < −5 and D2

2 > −5 + 2 length(Z) .

Moreover, since D2 6= 0, we have χ(X,OX(D2)) ≤ 0, so that D22 ≤ −4. It follows

that length(Z) = 0, and we have an exact sequence

0→ OX(D1)→ E → OX(D2)→ 0 ,

with D1, D2 of degree 0 and D21 = −6, D2

2 = −4. Then, D2 +H is a nodal curveof degree 2, which is a contradiction.

Proposition 4.35 now takes the form

Proposition 4.39. If X is a reflexive K3 surface with polarization H, and v =(2, `,−3), then there is a locally free rank 2 universal sheaf Q on X ×MH(v) →MH(v) making MH(v) a fine moduli scheme parameterizing locally free µ-stablesheaves with vector v. Moreover, X = MH(v) is a projective K3 surface and theintegral functor ΦQ

X→X is a Fourier-Mukai transform.


We start now an analysis which will culminate in the proof that the K3surface X is itself a fine moduli space of µ-stable bundles on X. Henceforth weassume that X is strongly reflexive.

Let E = Qy for a point y ∈ X. Since H2(X, E(H)) = 0 and χ(E(H)) = 1,the sheaf E(H) has at least one section.

Lemma 4.40. For every section of E(H) there is an exact sequence

0→ OX → E(H)→ Ix(`+ 2H)→ 0 .

Moreover, dimH0(X, E(H)) = 1, so that the point x depends only on the sheaf E,and dim Ext1(Ix(`+ 2H),OX) = 1.

Proof. Given a section of E(H), we have an exact sequence 0 → OX → E(H) →K → 0. By taking double duals, we obtain the exact sequence

0→ OX(D)→ E(H)→ IZ ⊗OX(`+ 2H −D)→ 0 , (4.9)

where Z is a zero-dimensional subscheme and D is an effective divisor of degree1, so that D2 ≥ −2. Moreover, H − 2D 6≡ 0, so that Hodge index theorem impliesthat 4D2 − 2 < 0 and we have two cases, D2 = 0 and D2 = −2. If D2 = 0, then(H − 2D)2 = −2, so that either H − 2D or 2D −H is effective, which is absurd.Thus, D2 = −2, and D is a nodal curve of degree 1, a situation we are excluding.Hence D = 0 and length(Z) = 1.

As far as the second statement is concerned, since H0(X,O(` + 2H)) =0, one has H0(X, Ix(` + 2H)) = 0 so that from the sequence (4.9) we obtaindimH0(X, E(H)) = 1. Moreover, by Riemann-Roch we have Hi(X,O(`+ 2H)) =0 also for i = 1, 2, so that Serre duality gives dim Ext1(Ix(` + 2H),OX) =dimH1(X, Ix(`+ 2H)) = 1.

Corollary 4.41. The line bundle OX(H) is IT0 and OX(H) ' OX(−ˆ− H), where−ˆ= γ0,2 is the (0,2) Kunneth part of the Chern character chQ.

Proof. By Lemma 4.40, OX(H) is IT0 and its Fourier-Mukai transform is a linebundle N . The sheaf N is identified by computing its first Chern class by Grothen-dieck-Riemann-Roch.

Lemma 4.40 implies that there is a one-to-one set-theoretic map Ψ: X → X,given by Ψ(y) = x, where x is the point determined by E = Qy.

Proposition 4.42. The map Ψ is an isomorphism of schemes.


Proof. The natural morphism π∗XN → Q⊗π∗XOX(H) whereN = OX(H) provides

a section0→ OX×X

τ→ Q⊗ π∗XOX(H)⊗ π∗XN−1 → K → 0 .

Let j : Z → X×X be the closed subscheme of zeroes of τ , and let p = πj : Z → X,p = π j : Z → X be the proper morphisms induced by the projections πX andπX .

It is enough to show that the morphism p : Z ∼→ X is an isomorphism ofschemes and the map Ψ is the composite morphism p p−1 : X → X. Now, oneeasily sees that for every (closed) point y ∈ X, τ induces a section 0 → OX

τy→E(H)→ Ky → 0 of E(H). By Lemma 4.40, Ky ' Ip(y)(`+ 2H) for a well-definedpoint p(y) ∈ X. Then, every closed fiber of x : Z → X consists of a single point andx is a proper finite epimorphism of degree 1 by Zariski’s main theorem. Since X issmooth, p is an isomorphism. Moreover one has Ψ = p p−1. As a consequence ofLemma 4.40, for every (closed) point y ∈ X the fiber Ψ−1(Ψ(y)) is a single point.If Ψ(X) is the scheme-theoretic image of Ψ, X → Ψ(X) is a finite epimorphismof degree 1 as above, so that dim Ψ(X) = 2 and Ψ(X) = X. The smoothness ofX yields the result.

Corollary 4.43. Let E be a sheaf which fits into an exact sequence

0→ OX → E(H)→ Ix(`+ 2H)→ 0 ,

where Ix is the ideal sheaf of a point x ∈ X. Then E is µ-stable and locally freewith v(E) = v = (2, `,−3) so that it defines a point y ∈ X and Ψ(y) = x.

Proof. According to Proposition 4.39, it is enough to prove that E(H) is stable,which is easily checked.

Lemma 4.40 suggests that the universal bundle Q can be obtained as anextension of suitable torsion-free rank 1 sheaves on X × X. Let IΨ be the idealsheaf of the graph ΓΨ : X → X × X of Ψ.

Lemma 4.44. The direct image πX∗[Ext1(IΨ ⊗ π∗XOX(` + 2H),OX×X)] is a line

bundle L on X.

Proof. Write E = `+ 2H and OΨ = (ΓΨ)∗OX . Then,

Ext1(IΨ ⊗ π∗XOX(E),OX×X) ' OΨ ⊗ π∗XOX(−E) .

By Lemma 1, RiπX∗π∗XOX(−E) = 0 for i ≥ 0, hence, from the exact sequence

0→ IΨ ⊗ π∗XOX(−E)→ π∗XOX(−E)→ OΨ ⊗ π∗XOX(−E)→ 0 ,


we obtain πX∗(OΨ ⊗ π∗XOX(−E)) ∼→ R1πX∗(IΨ ⊗ π∗XOX(−E)). But for everyy ∈ X, one has H1(X, IΨ⊗π∗XOX(−E)⊗Oy = H1(X, Ix(−E)), where x = Ψ(y),and one concludes by Lemma 4.29 and by Grauert’s cohomology base changetheorem.

It follows that the sheaf Ext1(IΨ ⊗ π∗XOX(`+ 2H), π∗X

(L−1)) has a section,so that there is an extension

0→ π∗X

(L−1)→ P → IΨ ⊗ π∗XOX(`+ 2H)→ 0 . (4.10)

Moreover, Lemma 4.40 implies that P⊗π∗XOX(−H) is a universal sheaf on X×X;thus P ∼→ Q⊗ π∗

XG ⊗ π∗XOX(H) for a line bundle G on X. The sheaves L and G

are readily determined; by applying πX∗ to the sequence above, one obtains

L−1 = OX(H)⊗ G ∼→ OX(−ˆ− H)⊗ G .

Now, by restricting the exact sequence (4.10) to a fiber π−1(x), we obtain c1(G) =−H. Then we have

πX∗(OΨ ⊗ π∗XOX(−E)) ∼→ OX(ˆ+ 2H) (4.11)

and

Proposition 4.45. The sequence of coherent sheaves on X × X

0→ π∗XOX(−ˆ−2H)→ Q⊗π∗

XOX(−H)⊗π∗XOX(H)→ IΨ⊗π∗XOX(`+2H)→ 0

is exact.

This implies that there is an exact sequence

0→ OX → F(H)→ Iy(ˆ+ 2H)→ 0

so that dimH0(X,Q∗x(H)) = 1 and dim Ext1(Iy(ˆ+ 2H),OX) = 1.

Proposition 4.45 allows us to compute the Chern character γ = chQ of Q.In particular, we obtain that the (2, 2) Kunneth part of γ is

γ2,2 = (`+ 2H) ∪ H +H ∪ ˆ− ι ,

where ι ∈ H2(X,Z)⊗H2(X,Z) is the element corresponding to the isometry Ψ∗ :H2(X,Z) → H2(X,Z). Taking into account that H = −πX∗(γ2,2H), one hasΨ∗(H) = 5H + 2ˆ. Moreover, Equation (4.11) gives Ψ∗(−`− 2H) = ˆ+ 2H. Fromthis we get the symmetric relations

Ψ∗(H) = 2ˆ+ 5H Ψ∗(`) = −5ˆ− 12H

H = Ψ∗(2`+ 5H) ˆ= Ψ∗(−5`− 12H) (4.12)

We need to show that the sheaves Q∗x are µ-stable with respect to H. A firststep is the following.


Proposition 4.46. The K3 surface X, equipped with the divisors H and ˆ, is re-flexive.

Proof. By Equation (4.12) we have ˆ2 = −12 and ˆ · H = 0. Moreover, sinceˆ + 2H = −Ψ∗(` + 2H), this divisor has negative degree with respect to thepolarization Ψ∗H, so that it is not effective.

In order to prove that X is a moduli space of stable sheaves on X, the mostnatural thing to do would be to use Corollary 4.43. However we cannot do thatbecause we do not know a priori if the reflexive K3 surface X is strongly reflexive.This problem is circumvented as follows.

Proposition 4.35 implies that the moduli space MH(v) of stable sheaves on X(with respect to H) with Mukai vector v is nonempty and connected and consistsof locally free sheaves.

Actually any sheaf F in MH(v) is µ-stable. Otherwise, proceeding as in theproof of Proposition 4.38, one could see that it can be destabilized by a sequence

0→ OX(D1)→ F → OX(D2)→ 0 ,

with D1, D2 of degree 0 with respect to H and D21 = −6, D2

2 = −4, D1 ·D2 = −1.Then D2 + H is a nodal curve and (D2 + H) ·Ψ∗H = 0, which is absurd becauseΨ∗H is ample. Thus F is µ-stable.

We may now proceed as in the proof Lemma 4.40; the sheaf F fits into anexact sequence

0→ OX → F(H)→ Iy(ˆ+ 2H)→ 0

for a well-defined point y ∈ X unless F is given by an extension

0→ OX(D)→ F(H)→ IZ(ˆ+ 2H −D)→ 0

where D is a nodal curve with D · H = 1 and Z is a zero-dimensional closedsubscheme of X. In the latter case, Hi(X, IZ(ˆ + 2H − D)) = 0 for i ≥ 0 sothat length(Z) = 0 and −4 = (ˆ+ 2H −D)2. Then, D · ˆ = −3 and D · Ψ∗H =D · (5H + 2ˆ) = −1, which is absurd since Ψ∗H is ample. Then, one has

0→ OX → F(H)→ Iy(ˆ+ 2H)→ 0 ,

for a point y ∈ X, and F ' Q∗x = ΦQ∗

X→X(Ox), with x = Ψ(y), since dim Ext1(Iy(ˆ+2H),OX) = 1.

This implies that the Fourier-Mukai transform ΦQX→X , maps F to Ox. By

applying Proposition 2.63 this induces an immersion of schemes MH(v) → X.Since the two schemes are irreducible and have the same dimension, they areisomorphic. As a consequence, we eventually obtain the sought-for result.


Theorem 4.47. X is a fine moduli space of µ-stable bundles on X (polarized byH) with invariants (2, ˆ,−3), and the relevant universal sheaf is Q∗.

The main result we have so far obtained in this section is the following: if Xis a strongly reflexive K3 surface, we have realized X as a moduli space of locallyfree µ-stable sheaves on X itself. In this sense, one could say that strongly reflexiveK3 surfaces are “self-dual.”

We finish this section with the computation of the topological invariants ofthe Fourier-Mukai transform ΦQ

X→X(F•) of an object in Db(X) in terms of those ofF•. The formula is obtained by the Riemann-Roch theorem, taking into accountthat we can compute the Chern character γ of Q from Proposition 4.45.

Proposition 4.48. Let u = (ρ, c1, σ) = (rk(F•), c1(F•), rkF• + ch2(F•)) be theMukai vector of F•, and d = c1 ·H. If u = v(ΦQ

X→X(F•)) = (ρ, c1, σ), one has

ρ = −3ρ+ 2σ + ` · c1,

c1 = (` · c1 + 2d)H + (ρ+ d− s)ˆ−Ψ∗(c1),

σ = 2ρ− 3σ − ` · c1 .

Then χ(ΦQX→X(F•)) = −χ(F•).

We also recover that u2 = u2, something we already know since the map f

is an isometry.

Corollary 4.49. If F is a WITi sheaf on X, then χ(F) = (−1)i+1χ(F) and c1(F) ·H = (−1)i+1c1(F) · H. In particular, the Euler characteristic and the degree ofWIT1 sheaves are preserved.

4.3.3 Homogeneous bundles

In Section 3.3, we have considered homegeneous bundles on Abelian varieties,namely, bundles that are invariant under translations. These can also be char-acterized as the coherent sheaves whose Abelian Fourier-Mukai transform is askyscraper sheaf. By means of the Fourier-Mukai we have introduced in the pre-vious sections we can generalize this notion to the case of strongly reflexive K3surfaces.

So we consider a strongly reflexive K3 surface X and the universal bundle Qgiven by Proposition 4.54.

Lemma 4.50. If E is a µ-stable locally free sheaf of degree zero, and v(E∗) 6=(2, `,−3), then E is IT1. In particular, every zero-degree line bundle on X is IT1.If E is a µ-stable coherent nonlocally free sheaf of degree zero, then E is IT1 unlessthere is an exact sequence 0 → E → Q∗ξ → OZ → 0, for a point ξ ∈ X and azero-dimensional closed subscheme Z → X.


Proof. For every ξ ∈ X we have H2(X,F ⊗ Qξ)∗ ' Hom(Qξ,F∗). Since F andQξ are µ-stable, if there exists a nonzero morphism Qξ → F∗, then it is anisomorphism, which is incompatible with the condition in the statement. The sameargument also shows that H0(X,F ⊗ Qξ) ' Hom(F∗,Qξ) = 0, thus concludingthe proof of the first claim.

For the second claim, if E is a µ-stable nonlocally free sheaf of degree zero, forevery ξ ∈ X we have H2(X, E ⊗Qξ)∗ ∼→ Hom(E ,Q∗ξ). Since E and Q∗ξ are µ-stable,for any nonzero homomorphism f ∈ Hom(E ,Q∗ξ) there is an exact sequence

0→ E f→ Q∗ξ → K → 0

with rk(K) = 0. Moreover K has degree zero, and then K ∼→ OZ for a zero-dimensional closed subscheme Z, a situation we are excluding. Then H2(X, E ⊗Qξ)∗ = 0 for every ξ ∈ X. On the other hand, H0(X, E ⊗ Qξ)∗ ∼→ Hom(Q∗ξ , E). Iff ∈ Hom(Q∗ξ , E) is nonzero, we have as above

0→ Q∗ξf→ E → K′ → 0

with rk(K′) = 0. After dualizing we get again a contradiction. Then H0(X, E ⊗Qξ)∗ = 0 for every ξ ∈ X, thus concluding the proof.

Definition 4.51. A coherent torsion-free sheaf E on X is homogeneous if there isa filtration by coherent sheaves 0 = E0 ⊂ E1 ⊂ · · · ⊂ Es = E such that every sheaf(Ei+1/Ei)∗ is µ-stable with Mukai vector v, namely, it is a sheaf defining points ofX. 4

Note that homogeneous sheaves are µ-semistable (since the quotients of theirfiltration are µ-stable).

An analogous definition applies for sheaves on X, looking at X as a mod-uli space of stable sheaves on X with universal bundle Q∗. These homogeneoussheaves play, in a sense, the same role as homogeneous sheaves on Abelian surfacesdescribed in Section 3.3. In that case, sheaves that admit a filtration by line bun-dles of zero degree are just homogeneous sheaves, that is, sheaves invariant undertranslations by points of the Abelian surface. The relevance of this definition isshown by the following result.

Proposition 4.52. If T is a coherent sheaf on X with zero-dimensional support, itis IT0, and its Fourier-Mukai transform T is homogeneous. Conversely, if E is ahomogeneous sheaf on X, then it is WIT2, and its Fourier-Mukai transform E haszero-dimensional support.

Proof. The second statement is the dual of the first one. We prove the first state-ment by induction on the length m of the support of T . For m = 1, this reduces


to the statement that ΦQX→X(Ox) ' Qx, which we already know. Otherwise, if p is

a point of the support, there is an exact sequence

0→ T ′ → T → Op → 0 ,

where T ′ has a zero-dimensional support of length m−1. By the inductive hypoth-esis, T ′ is IT0 and T ′ is quasi-homogeneous. From the previous exact sequence,we see that T is IT0, and that there is an exact sequence

0→ T ′ → T → Op → 0 ,

which implies that T is quasi-homogeneous as well.

Corollary 4.53. Let E be a µ-stable sheaf of degree zero. If E is locally free, it iseither WIT2 or IT1 according to whether v(E∗) = (2, `,−3) or not. If E is neitherlocally free nor IT1, there is an exact sequence

0→ E → Q∗ξ → OZ → 0 ,

for a point ξ ∈ X and a zero-dimensional closed subscheme Z → X, so thatΦ0(E) = 0, Φ1(E) is quasi-homogeneous and Φ2(E) ' Oξ, where Φ = ΦQ

X→X.

4.3.4 Other Fourier-Mukai transforms on K3 surfaces

Mukai’s construction

Strongly reflexive K3 surfaces have been the first example of a class of K3 surfacessupporting a Fourier-Mukai transform. Another example was provided by Mukai[228]. The description we give here of that example is quite different from theoriginal treatment by Mukai since we can take advantage of the results proved inChapter 1 and in Section 4.1.

Let X be an algebraic K3 surface, and assume there exist coprime positiveintegers r, s and a polarization H in X such that H2 = 2rs. Let us consider themoduli space X = MH(r,H, s) of stable sheaves on (X,H), with Mukai vectorv = (r,H, s). Note that v2 = 0. The moduli space X = MH(r,H, s) is nonemptyas a consequence of the following result.

Proposition 4.54. If (X,H) is a polarized K3 surface, and v = v(r,H, s) is aprimitive Mukai vector with v2 = 0, the moduli space X = MH(v) is nonempty.Moreover, X is a K3 surface, and there is a universal family Q on X × X.

Proof. The first claim is [227, Thm. 5.4]. To prove the second, note that thegreatest common divisor of the numbers r, H2 = 2rs and s is 1 since r and s arecoprime. As a consequence the moduli space X is compact. Again because r ands are coprime, Theorem 4.20 applies, and a universal family on X × X exists. ByTheorem 4.25, X is a K3 surface.


Remark 4.55. Note that by Proposition 4.21, both integral functors ΦQX→X and

ΦQX→X are Fourier-Mukai transforms. 4

Let M be the lattice having generators e, f with intersection numbers

e2 = −2r, e · f = s+ 1, f2 = 0 .

By the surjectivity of the period map [22, Chap. 8], there exists a K3 surface Xwith Picard lattice isomorphic to M (cf. also [93]).

Lemma 4.56. The class h = e+ rf ∈ Pic(X) is ample.

Proof. The Grothendieck-Riemann-Roch formula shows that, possibly after re-placing the pair e, f with the pair −e,−f, the class f may be assumed to beeffective. Moreover, the generic element in the linear system |f | is an irreducibleelliptic curve. Let C be the class of an irreducible curve, and let C = me + nf ;then, m ≥ 0 as C · f = ms. Notice that h · C = ns. Since C2 ≥ −2, we have

ms(n−mr) ≥ −1. (4.13)

If m > 0, we have n > 0 by (4.13), while if m = 0, C = nf implies againn > 0. Therefore h · C > 0, and since also h2 = 2rs > 0, h is ample by Nakai’scriterion.

Notice that in this K3 surface X there are divisors of degree 2s (for instance,D = 2f). The previous results apply to the K3 surface X (note in particular thath2 = 2rs), so we may consider the moduli space X = Mh(r, h, s) of stable sheaveson X. Then for every m ∈ Z, the complex ΦQ

X→X(OX(mh−D)) has rank zero. Asa consequence, the determinant bundle

Lm = (det ΦQX→X(OX(mh−D))−1 ⊗ (det ΦQ

X→X(OX(−mh−D))

is independent of the choice of the universal family. Moreover, for m big enough,Lm is ample [101].

Let us define a divisor class h in X by letting

h = −µ(h) + 2sφ ,

where µ is the morphism defined in Equation (4.6) and φ is the Kunneth (0,2) partof the Chern class c1(Q). A direct computation by the Grothendieck-Riemann-Roch theorem yields

Lm = 2mh ,

so that h is ample.


Now, following Mukai [228], and in analogy with the result in Proposition4.45, we construct a morphism X → X. Let G the sheaf on X associated with thepresheaf

U Ext1X×U (I∆ ⊗ π∗1OX(e+ f), π∗1OX(f))

where I∆ is the ideal sheaf of the diagonal ∆ ⊂ X × X. The morphisms π1,π2 are here the projections onto the factors of X × X. Since Hom(Ix(e),OX) =Ext2(Ix(e),OX) = 0 for every x ∈ X, the sheaf G is locally free, and G ⊗ k(x) 'Ext1(Ix(e),OX). Then there is a coherent sheaf E on X ×X fitting into an exactsequence

0→ π∗1OX(f)⊗ π∗2G∗ → E → I∆ ⊗ π∗1OX(e+ f)→ 0 .

For every x ∈ X, let Ex = E|X×x. Note that Ex fits into the exact sequence

0→ OX(f)⊕(r−1) → Ex → Ix(e+ f)→ 0 . (4.14)

One has v(Ex) = (r, h, s). Moreover,

Proposition 4.57. For every x ∈ X, the sheaf Ex is locally free and h-stable.

Proof. Starting for the exact sequence (4.14) one proves that Ex ' (Ex)∗∗, so thatEx is locally free. To prove the second claim, let F be a proper torsion-free quotientof Ex(−f) and let F0 be the image of the composition O⊕(r−1)

X → Ex(−f) → F ,where the first arrow is the morphism in the sequence (4.14). We have a diagram

0

0

0

0 // K1

// K2

// K3

// 0

0 // O⊕(r−1)X

// Ex(−f)

// Ix(e)

// 0

0 // F0

// F

// F ′′

// 0

0 0 0

The sheaf K3 either has rank 1, or is zero. In the second case, rk(F0) = rk(F)− 1,and c1(F) = c1(F0) + e, and

µ(F) ≥ h · erk(F)

>h · er

= µ(Ex(−f)) ,


so that Ex is stable (note that deg(F0) ≥ 0 since F0 is a quotient of O⊕(r−1)X ).

If rk(K3) = 1, then rk(F0) = rk(F). Now, let us notice that det(F0) iseffective, since by taking the ρ-th exterior power of the morphism O⊕(r−1)

X → F0

in the first column of the previous diagram, one obtains a nonzero morphism

O⊕N → det(F0) (here ρ = rk(F0), and N =(r − 1ρ

)).

Let c1(F0) = me+ nf . Due to Lemma 4.59, one has

µ(F) ≥ µ(F0) =h · (me+ nf)

rk(F)≥ m(rs− 1)

rk(F)≥ ms .

Now, if m ≥ 1 thenµ(F) > s− 1 = µ(Ex(−f))

so that Ex is stable. So let m = 0. As we have already noticed, that the genericmember in the linear system |f | is an irreducible elliptic curve, hence h0(OX(nf)) =n + 1. In view of Lemma 4.60 there is a morphism O⊕(ρ+1)

X → F0 which is sur-jective out of a finite sets of points. The kernel of this morphism is isomorphic to(det(F0))−1, so that we obtain an exact sequence

0→ (det(F0))−1 → O⊕(ρ+1)X → F ′0 → 0 .

The associated cohomology long exact sequence contains the segment

0→ H1(X,F ′0)→ H2(X, (det(F0))−1)→ H2(X,O⊕(ρ+1)X )→ H2(X,F ′0)→ 0 .

We may assume that F is stable, and then F∗0 is stable as well. Since deg(F∗0 ) ≤ 0we have H0(X,F∗0 ) ' H2(X,F0) = 0. As F0/F ′0 is supported on points, we alsohave H2(X,F ′0) = 0. As a consequence, n+ 1 = h0(det(F0)) ≥ ρ+ 1, so that

µ(F) ≥ µ(F0) =h · nfrk(F)

≥ f · h = s+ 1 > µ(Ex(−f)) .

So we have a “classification morphism” Ψ, mapping x ∈ X to the sheaf Exin X = Mh(r, h, s).

Corollary 4.58. The classification morphism Ψ is an isomorphism.

Proof. It is enough to prove that Ex ' Ey implies x = y. Indeed if this is thecase, Ψ is an open embedding, and since X is connected, Ψ is an isomorphism.To prove this claim, let us note that an isomorphism Ex ∼→ Ey maps the subsheafO⊕(r−1)X ⊂ Ex to the subsheaf O⊕(r−1)

X ⊂ Ey, since Hom(OX ,OX(e)) = 0. ThenIx ' Iy, and x = y.


We prove now the two lemmas that have been used to prove Proposition 4.57.

Lemma 4.59. If a divisor class D = me+ nf is effective, then n ≥ 0 and h ·D ≥(rs− 1)m.

Proof. The proof is a simple computation. See [228], Lemma 3.3.

Lemma 4.60. Let X be a smooth projective surface X such that H1(X,OX) = 0.Let G be a sheaf on X whose torsion is supported on points, and assume that thereis a morphism V ⊗OX → G (where V is a finite-dimensional vector space) whosecokernel is supported on points. Then there is a morphism V ′ ⊗ OX → G (whereV ′ is a vector space of dimension rk(G)+1) whose cokernel is supported at a finiteset of points.

Proof. We prove this result first for rk(G) = 1 and then extend it by induction.We may assume that dimV ≥ 2. Composing the morphism V ⊗ OX → G withthe natural morphism G → G∗∗, we obtain a morphism V ⊗ OX → G∗∗ whosecokernel is supported on points, since the torsion of G is supported on points. Thismeans that there exist s1, s2 ∈ V whose corresponding divisors, when they areregarded as sections of the line bundle G∗∗, intersect at finite number of points. IfV ′ = 〈s1, s2〉, the cokernel of the morphism V ′ ⊗OX → G is supported on points.

To trigger the induction mechanism we need to show that if rk(G) ≥ 2, ageneric element of s ∈ V induces an exact sequence 0→ OX

s−→ G → G′ → 0 suchthat the torsion of G′ is supported on points. To prove this, let U ⊂ X be the opensubset where G is locally free and the morphism V ⊗OX → G is surjective (underour hypotheses, the complement of U is a finite set of points). By [119, Example12.1.11], for a generic s ∈ V the zero locus Z of s|U is empty or is a finite numberof points. Then G′ is locally free on U − Z.

Now we can draw a commutative diagram

0 // OX // V ⊗OX

// W ⊗OX

// 0

0 // OXs // G // G′ // 0

where W and V are vector spaces of dimension rk(G) and rk(G) + 1, respectively,and the cokernel Q′ of the rightmost vertical arrow is supported on points bythe induction hypothesis. The morphism V ⊗OX → G exists because a morphismOX → G′ can be lifted to a morphism OX → G since H1(X,OX) = 0. The cokernelof the morphism V ⊗OX → G is isomorphic to Q′.

In this way, we have constructed another example of a family of K3 surfacesthat are “self-dual,” i.e., each of them is isomorphic to a component of the moduli


space of stable bundles on it, having suitable topological invariants. Moreover,these components of the moduli space are fine, and the corresponding universalsheaves define Fourier-Mukai transforms (cf. Remark 4.55).

A Fourier-Mukai transform by extension of the kernel

It is not difficult to construct new Fourier-Mukai transforms out of given ones, forinstance by taking extensions of the kernels. We give here an example based onstrongly reflexive K3 surfaces (taken from [73] with some changes).

The normalization Rπ∗Q ' OX [−1] implies that dimH1(X × X,Q) = 1,as the Leray spectral sequence shows immediately. So there is a unique nontrivialextension

0→ Q→ U → OX×X → 0 . (4.15)

Proposition 4.61. The restrictions of the sheaf U to the varieties X × ξ, withξ ∈ X, and p × X, with p ∈ X, are all stable.

Proof. We consider only the second type of restriction, since the proof is thesame in the two cases. The sheaf Uξ = UX×ξ is µ-semistable with vanishingdegree. Let F be a destabilizing proper subsheaf of Uξ, which we may assumeto be stable. Then χ(F)/ rk(F) ≥ χ(Uξ)/rk(Uξ) = 1

3 . Let f : F → OX be thecomposite morphism. One necessarily has f 6= 0, otherwise there would be anonzero morphism F → Qξ and then χ(F)/ rk(F) ≤ − 1

2 , which contradicts theprevious inequality. But then then F ' OX , which implies in turn Uξ ' OX ⊕Qξ.

From this we get

Φ0(Uξ) = 0 , Φ1(Uξ) ' OX , Φ2(Uξ) ' Oξ .

But then the spectral sequence (2.35) (or, to be more precise, the spectral sequenceassociated to the composition Φ Φ) degenerates at the second step, yielding acontradiction.

By the general theory this implies that the kernel U gives rise to a Fourier-Mukai transform Ψ = ΦU

X→X . Let Ψ = ΦU∗

X→X be the inverse transform. The trans-form Ψ has some nice features.

Proposition 4.62. The Fourier-Mukai transform Ψ and its inverse Ψ satisfy thefollowing properties.

1. Ψ(OX) ' OX [−2].

2. Ψ(Iξ) ' Q∗ξ [−1] for all ξ ∈ X.

4.4. Preservation of stability 139

Proof. The first claim is proved by inspection of the long exact sequence that oneobtains by applying the functor Rπ∗ to the sequence (4.8). To prove the secondclaim note that Ψ(OX) ' OX by the previous result, and that Ψ(Oξ) ' U∗ξ . Thenby applying Ψ to the exact sequence 0→ Iξ → OX → Oξ → 0, we obtain

0→ OX → U∗ξ → Ψ1(Iξ)→ 0

which proves the claim.

The second property implies that the ideal sheaf IZ of a zero-cycle Z in X

is IT1 with respect to Ψ and its transform is the dual of the quasi-homogeneousbundle on X corresponding to Z.

4.4 Preservation of stability

In this section we study the behavior of µ-(semi)stable sheaves on reflexive K3 sur-faces under the Fourier-Mukai transform. Our treatment is inspired by techniquesdeveloped by Maciocia for Abelian surfaces [203] already described in Section 3.5.2.Another approach which involves transcendental techniques will be developed inChapter 5.

Lemma 4.63. Let E be a coherent sheaf on X.

1. If E is IT0, then deg(E) ≥ 0, and deg(E) = 0 if and only if E has zero-dimensional support.

2. If E is WIT2, then deg(E) ≤ 0, and deg(E) = 0 if and only if E is a quasi-homogeneous sheaf.

Proof. We recall that deg(E) = (−1)i+1 deg(E) for a WITi sheaf by Corollary 4.49.

1. Let us consider the exact sequence

0→ T → E → F → 0 , (4.16)

where T is the torsion subsheaf of E . Then deg(T ) ≥ 0 so that deg(F) ≤ deg(E)with equality if and only if T is supported in dimension zero. Suppose thatdeg(E) ≤ 0; then deg(F) ≤ 0. Since T has rank zero, H2(X, T ⊗Qξ) = 0 for everyξ ∈ X, so that Φ2(T ) = 0 and the exact sequence (1.10) applied to (4.16) provesthat F is IT0. Then F is WIT2 and locally free of degree deg(F) = −deg(F) ≥ 0.This implies that

Hom(F ,Qp) ∼→ H2(X, F ⊗ Q∗p)∗ 6= 0

for every point p ∈ X, so that there exists a nonzero morphism F → Qp.


We prove that F is µ-stable. Suppose indeed that there is a destabilizingsequence

0→ L → F →M→ 0 , (4.17)

where L is a stable locally free sheaf with µ(L) ≥ µ(F) ≥ 0. It is enough to seethat Φ2(L) = 0 because in this case L is WIT1 and Φ0(M) ' Φ1(L), which isabsurd since Φ0(M) is WIT2 and Φ1(L) is WIT1. If Φ2(L) 6= 0, there exists apoint p ∈ X such that H2(X,L ⊗ Q∗p) 6= 0, so that there is a nonzero morphismf : L → Qp. This implies µ(L) ≤ µ(Qp) = 0, and then µ(L) = 0 and f is anisomorphism. Then Φ2(L) ' Op, and applying Φ to the exact sequence (4.17) weget Φ0(M) = 0 and 0→ Φ1(M)→ Op → F → Φ2(M)→ 0. Thus Φ1(M) ' Op,because F is torsion-free. We now consider the spectral sequence given by Equation(4.3); one has Ep,q2 = ΦpΦq(M), E2

∞ ' M, and Ep+q∞ = 0 for p + q 6= 2. SinceE2,0

2 = 0, any nonzero element in E0,12 ' Φ0(Op) ' Qp is a cycle which survives

to infinity; thus E1∞ 6= 0, which is absurd.

Since F and Qp are both µ-stable, the existence of a nonzero morphismF → Qp implies that µ(F) = 0. Then deg(F) = 0, so that deg(T ) = 0; as aconsequence, T has zero-dimensional support, and the same is true for the sheafE .

2. If E is WIT2, then E is IT0, so that deg(E) ≥ 0 with equality if and onlyif E has zero-dimensional support, by 1 on X. Hence, by Proposition 4.52 we havedeg(E) ≤ 0 with equality if and only if E is quasi-homogeneous.

Proposition 4.64. Let E be a coherent sheaf on X. If E is µ-semistable and WIT1

with deg(E) = 0, then its Fourier-Mukai transform E is µ-semistable.

Proof. Since deg(E) = 0, if E is not µ-semistable, there is an exact sequence

0→ F → E → G → 0 , (4.18)

where G is torsion-free and deg(F) ≥ 0 ≥ deg(G). In general, we do not getstrict inequalities because we cannot assume that E is torsion-free; the equalitiesdeg(F) = 0 = deg(G) hold only if E has torsion and F has zero-dimensionalsupport. The long exact sequence 1.10 applied to (4.18) gives Φ0(F) = Φ2(G) = 0and exact sequences

0→ Φ0(G)→ Φ1(F)→ K → 0 , 0→ K → E → Φ1(G)→ Φ2(F)→ 0 .

In particular, F is not IT0 and, by Proposition 4.52, its support is not zero-dimensional, whence deg(F) > 0. Thus E is torsion-free.

Now, Φ0(G) is WIT2 and Φ2(F) is IT0 by Corollary 4.24, so that Lemma4.63 implies deg(Φ0(G)) ≤ 0 and deg(Φ2(F)) ≥ 0. Then deg(Φ1(F)) = deg(F) +deg(Φ2(F)) > 0, so that deg(K) = deg(Φ1(F))−deg(Φ0(G)) > 0, thus contradict-ing the semistability of E .

4.4. Preservation of stability 141

Lemma 4.65. If F is locally free and T has zero-dimensional support, every exactsequence 0→ F → K → T → 0 splits.

Proof. By local duality the sheaves ExtiOX (T ,F) vanish for i 6= 2. Then Ext1(T ,F)= 0 and the exact sequence splits.

Corollary 4.53 characterizes the µ-stable IT1 sheaves of degree zero on X.The next proposition and its corollaries study the preservation of stability of suchsheaves.

Proposition 4.66. Let E be a coherent µ-stable IT1 locally free sheaf of degree zeroon X. The Fourier-Mukai transform E is µ-stable.

Proof. Since E is locally free, if it is not µ-stable it can be destabilized by asequence

0→ F → E → G → 0 ,

where F is µ-stable and locally free, rk(F) < rk(E) and G is torsion-free.

We have two cases:

(a) F is IT1. We have Φ2(G) = 0 and two exact sequences

0→ Φ0(G)→ F → K → 0 , 0→ K → E → Φ1(G)→ 0 . (4.19)

Again, Φ0(G) is WIT2 and deg(Φ0(G)) ≤ 0, so that

deg(K) = deg(F)− deg(Φ0(G)) ≥ 0 .

Since E is µ-stable, the only possibility is deg(K) = 0 and rk(K) = rk(E), so thatrk(Φ1(G)) = 0. Since deg(Φ1(G)) = 0, Φ1(G) has zero-dimensional support, henceit is IT0 by Proposition 4.52. On the other hand, deg(K) = 0 implies deg(Φ0(G)) =0, and then Φ0(G) is quasi-homogeneous, by Lemma 4.63 again. The first sequencein (4.19) shows that K is WIT1 and induces the exact sequence

0→ F → K → Φ2(Φ0(G))→ 0 .

By Proposition 4.52, Φ2(Φ0(G)) has zero-dimensional support, and since F islocally free and K is torsion-free, Lemma 4.65 forces Φ0(G) = 0, and then G isWIT1. But then Φ1(G) is WIT1 as well, and, since we have proved that it is IT0,we have G = 0, which is absurd as rk(F) < rk(E).

(b) F is not IT1. Then, by Corollary 4.53 and Proposition 4.52, F is WIT2

and F ∼→ κ(p) for a closed point p ∈ X. The long exact sequence (1.10) shows thatG is WIT1 and yields the exact sequence

0→ E → Φ1(G)→ κ(p)→ 0 .

This sequence splits by Lemma 4.65 , which is absurd since Φ1(G) is WIT1 andE ⊕ κ(p) is not.


Corollary 4.67. Let E be a IT1 torsion-free sheaf of degree zero on X. If E isstrictly µ-semistable, then its Fourier-Mukai transform E is strictly µ-semistable.

Proof. E is µ-semistable by Proposition 4.64. Since E is not µ-stable, Proposition4.66 for X implies that if E is µ-stable it cannot be IT1. Then by Corollary 4.53E is WIT2, which is absurd.

Corollary 4.68. If E is a nonlocally free IT1 µ-stable torsion-free sheaf of degreezero on X, its transform E is not µ-stable.

Proof. The sheaf E is locally free and WIT1. If it is µ-stable, by Corollary 4.53 itis IT1. But this contradicts the fact that E is not locally free.

The moduli space MH(v) of H-stable bundles on a K3 (or Abelian) surface Xwith Mukai vector v carries a naturally defined symplectic holomorphic structure,defined as follows. If [F ] is a point in MH(v), then tangent space T[F ](MH(v))may be identified with the vector space Ext1(F ,F). The cup product provides askew-symmetric map Ext1(F ,F) ⊗ Ext1(F ,F) → Ext2(F ,F). Moreover one hasa trace morphism Ext2(F ,F) → Ext2(OX ,OX) dual to the natural morphismHom(OX ,OX) → Hom(F ,F). Since Ext2(OX ,OX) ' C, this defines a holo-morphic 2-form on MH(v) which turns out to be closed and nondegenerate, thusdefining a holomorphic symplectic 2-form.

Remark 4.69. In Section 3.5.3, we introduced symplectic structures of modulispaces of sheaves on surfaces in terms of the Yoneda pairing on the Ext groups.Since the Fourier-Mukai transform is well behaved with respect to these groups,it should induce symplectomorphisms between the moduli spaces, as we indeedwere able to prove in the case of Abelian surfaces. The same happens in the caseof (strongly reflexive) K3 surfaces. Let X be an H-polarized strongly reflexive K3surface, fix a Mukai vector v, and let Mµ

H(v) be the subset of MH(v) formed bylocally free µ-stable sheaves on X. We also assume that v = (r, c, s) is such that chas degree zero, c ·H = 0, and Mµ

H(v) is nonempty. All sheaves parameterized bypoints in Mµ

H(v) are IT1 with respect to the Fourier-Mukai transform ΦQX→X . By

Proposition 4.66, the Fourier-Mukai transform defines a morphism ς : MµH(v) →

Mµ

H(v), where the Mukai vector v ∈ H•(X,Z) is given by Proposition 4.48. Again,

one proves that ς is symplectic. 4

4.5 Hilbert schemes of points on reflexive K3 surfaces

As a geometric application of the Fourier-Mukai transform on strongly reflexiveK3 surfaces, we show that the Hilbert scheme Hilbn(X) of such a surface X (pa-rameterizing zero-dimensional subschemes of X of length n) may be realized as a

4.5. Hilbert schemes of points on reflexive K3 surfaces 143

moduli space of stable bundles on X. For the definition of the Hilbert scheme see[135], or [155] for a more modern and readable account. Since X is a projectivesmooth surface, Hilbn(X) is smooth and projective as well.

We shall prove the following result.

Theorem 4.70. For any n ≥ 1, the Hilbert scheme Hilbn(X) is isomorphic to themoduli space Mn = MH(1 + 2n,−nˆ, 1 − 3n) of H-semistable sheaves on X withMukai vector (1 + 2n,−nˆ, 1− 3n).

As a matter of fact, one sees that all points of Mn correspond to stable locallyfree sheaves. The result established by Theorem 4.70 can be compared with severalresults about the birationality of the Hilbert scheme of points of a polarized K3surface (X,H) with a moduli space of H-stable sheaves of X. For instance, in [297]a birational map MH(2, 0,−1− n2H2)→ Hilb2n2H2+3(X) is constructed.

Now we prove Theorem 4.70. We shall follow [73]. Let Z be a zero-cycle inX. The standard tricks show that the ideal sheaf IZ is IT1, so that by applyingthe Fourier-Mukai transform to the sequence 0→ IZ → OX → OZ → 0 we get

0→ OZ → IZ → O bX → 0. (4.20)

Lemma 4.71. The FM transform IZ is stable.

Proof. Since ch(IZ) = (1, 0,−n), by the formulas in Proposition 4.48 we obtain

rk IZ = 1 + 2n, ch2 IZ = −5n ,

and moreover, denoting by P (IZ) = χ(IZ)/ rk IZ , we have

P (IZ) =2− n1 + 2n

> −12.

Let A be a destabilizing subsheaf of IZ , which we may assume to be stable witha torsion-free quotient. Then we have P (A) ≥ P (IZ) > −1

2 . Let f denote thecomposite A → IZ → OX .

There are two cases:

(i) f = 0. Then there is a map A → OZ . Let gk : A → Qpk be the compositionof this map with the canonical projection onto Qpk . Since P (A) > − 1

2 = P (Qpk)and both sheaves are stable, we obtain gk = 0 for all k, which is absurd.

(ii) f 6= 0. We divide this into two further cases: rkA = 1 and rkA > 1.

If rkA = 1 we have A∗ ' OX ; hence the sequence (4.20) splits, which

contradicts the inversion theorem IZ ' IZ .


If rkA > 1, we consider the exact sequences

0→ K1 → Ah−→ B → 0 and 0→ B → OX → K2 → 0,

where rkK2 = 0, 1. If rkK2 = 1 then B = 0, i.e., f = 0 which is absurd, so thatrkK2 = 0, and B has rank one. We have an exact commuting diagram

0 0 0

0 // K3//

OO

K4//

OO

K2//

OO

0

0 // OZ //

OO

IZ //

OO

OX //

OO

0

0 // K1//

g

OO

A //

OO

B //

h′

OO

0

0

OO

0

OO

0

OO

(4.21)

with µ(K1) = 0, 0 < rkK1 < 2n and f = h′ h.

If n = 1, then OZ is µ-stable, but this is a contradiction. For n > 1, we mayassume that K1 is µ-semistable so that it is a direct summand of OZ . Then K3

is locally free and rkK1 ≥ 2. Moreover, µ(B) ≤ 0 because B injects into OX , andµ(B) ≥ 0 because µ(K1) ≤ 0. Then µ(B) = µ(K1) = 0. Since K3 is locally free, thesupport of K2 is not zero-dimensional. So µ(B) = 0 implies K2 = 0 and K3 ' K4.

Finally, we consider the middle column in (4.21)). The sheaf A has rankgreater than 2, and is stable, so that it is IT1. But IZ is WIT1 while K4 is WIT2.Then A ' IZ , but this is a contradiction.

Note that IZ is never µ-stable because (4.20) destabilizes it.

Let Mn be the moduli space MH(1 + 2n,−nˆ, 1 − 3n) of stable sheaves onX. The previous construction yields a map Hilbn(X) → Mn which is algebraicbecause Fourier-Mukai transform yields a natural isomorphism of moduli functorsand so gives rise to an isomorphism of (coarse or fine) moduli schemes. This mapis injective due to the inversion theorem of the Fourier-Mukai transform. We shallnow show that this map is surjective as well.

Lemma 4.72. Any element F ∈Mn is WIT1.

Proof. Since P (F) > −12 and P (Qp) = − 1

2 , there is no map F → Qp. This meansthat H2(X,F ⊗ Q∗p) = 0. We consider now nonzero morphisms Qp → F . Any


such map is injective; otherwise it would factorize through a rank 1 torsion-freesheaf B with µ(B) > 0 (because Qp is µ-stable) and µ(B) ≤ 0 (because F isµ-semistable), which is impossible. Then Qp is a locally free element of a Jordan-Holder filtration of F . Since any such filtration has only a finite number of terms,and the associated grading gr(F)∗∗ is unique, there is only a finite number of p’sgiving rise to nontrivial morphisms, i.e., Hom(Qp,F) ' H0(X,F ⊗Q∗p) does notvanish only for a finite set of points p. This suffices to prove that F is WIT1 dueto [227, Prop. 2.26].

Lemma 4.73. The Fourier-Mukai transform F of F is torsion-free.

Proof. Let T be the torsion subsheaf of F , so one has an exact sequence

0→ T → F → G → 0 . (4.22)

Since T is supported at most by a divisor, and F is WIT1, the sheaf T is WIT1

as well. Moreover deg(T ) ≥ 0. If deg T = 0, then T is IT0, i.e., T = 0.

Hence, we assume deg(T ) > 0. The rank 1 sheaf G is torsion-free and, byembedding it into its double dual, we see that it is IT1. Then, applying the inverseFourier-Mukai transform Φ to (4.22) we get 0 → T → F → G → 0. Since F isµ-semistable, we see that deg T = deg T ≤ 0, which is a contradiction.

Now the Chern character of F is (1, 0,−n), so that F is the ideal sheaf of azero-dimensional subscheme of X of length n. We have therefore shown that theFourier-Mukai transform surjects as a map HilbnX →Mn.

Altogether, this establishes Theorem 4.70. This theorem and its proof havesome immediate consequences which we can state in the following proposition,where n is any positive integer.

Proposition 4.74. Let X be a K3 surface, and let H, ` be divisors that make itstrongly reflexive. The moduli space Mn of H-stable sheaves on X with Cherncharacter (1 + 2n,−n`,−5n) is connected and projective. All its points correspondto locally free sheaves, and it contains no µ-stable sheaves.

Proof. Only the last claim has not yet been demonstrated. To prove it one uses thefact that any µ-semistable sheaf of the given Chern character admits a surjectionto OX and so fits into a sequence of the form (4.20).


More on preservation of stability. A more general theorem about preservationon µ-stability on K3 surfaces has been given in [154]. This result requires the


introduction of some categories naturaly attached to the choice of an ample divisoron X; this goes beyond the scope of this chapter, see however Remark D.28.

The preservation of twisted Gieseker stability under the Fourier-Mukai trans-form has been shown by Yoshioka also in the case of K3 surfaces [294, 295, 296].See also Section D.3.2.

Hilbert schemes. Our result about the isomorphism of some moduli spaces of stablesheaves on a K3 surface X, and Hilbert schemes of points of X, is a particularcase of general results about the birationality of such spaces proved by Zuo [297],Qin [255], Gottsche and Huybrechts [129] and O’Grady [238].

Fourier-Mukai partner K3 surfaces. In the examples of Fourier-Mukai functorsconstructed in this chapter, the two “partner” K3 surfaces turn out to be isomor-phic. This is not necessarily the case. This is discussed in some detail in Chapter7, especially Section 7.4.4.

K3 fibrations. The material presented in this chapter is relevant to study of relativeintegral functors on varieties that are fibered in K3 surfaces. There is not muchmathematical literature about this subject. In [71] Bridgeland and Maciocia showthat a relative moduli space for a variety fibered in K3 surfaces or Abelian surfacesis smooth if and only if it is fine. In [277] Thomas studies some moduli spaces ofstable sheaves on K3 fibrations.

Calabi-Yau varieties fibered in K3 surfaces have been on the other handquite extensively studied in the physical literature as compactification spaces forheterotic string theories; among the main contributions one can cite [183, 11, 151,125, 21, 174].

Chapter 5

Nahm transforms

Introduction

The original Nahm transform, i.e., a mechanism that starting from an instanton ona 4-dimensional flat torus produces an instanton on the dual torus, was introducedby Nahm in 1983 [230]. This construction was formalized by Schenk [263] andBraam and van Baal [57] in later years. Their descriptions show that the Nahmtransform is essentially an index-theoretic construction: given a vector bundle Eon flat torus X, equipped with an anti-self-dual connection ∇, one considers thedual torus X as a space parameterizing a family of Dirac operators twisted by ∇.Taking the index of this family yields, under suitable conditions, the instanton ∇on X.

Braam-van Baal and Schenk already hinted at a connection between theNahm and the Fourier-Mukai transforms. A first description of their relation wasgiven by Donaldson and Kronheimer [102]. From an abstract point of view, thebridge between the two constructions is provided by a relation between indexbundles and higher direct images, very much in the spirit of Illusie’s definition ofthe “analytical index” of a relative elliptic complex [158, Appendix II]. The factthat the Nahm transforms maps instantons to intantons then corresponds, via theHitchin-Kobayashi correspondence, to the fact that sometimes the Fourier-Mukaitransform preserves the condition of stability.

The purpose of this chapter is to embed Nahm’s construction into a moregeneral class of transforms, which we call Kahler Nahm transforms. This will allowus to compare in a precise manner the Nahm and Fourier-Mukai transforms. Afterthat we further develop the theory, introducing a special case of such transformswhen the manifolds involved have a hyperkahler structure. We consider a gener-

147Progress in Mathematics 276, DOI: 10.1007/b11801_5,C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics,


148 Chapter 5. Nahm transforms

alization of the notion of instanton (the quaternionic instantons) and prove thatthe “hyperkahler Fourier-Mukai transform” preserves the quaternionic instantoncondition.

The Nahm transform has been widely used to study instantons admittingsymmetries — e.g., instantons that are periodic in one or more directions. Weshall not cover here these applications, just restricting ourselves to provide somerelevant bibliography in the “Notes and further reading” section.

In the first section we provide the reader with some notions that will beneeded in the chapter — basically, the concept of instanton, a cursory view ofthe Hitchin-Kobayashi correspondence, and a review of Dirac operators and indexbundles.

5.1 Basic notions

5.1.1 Connections

In this section we shall consider complex vector bundles E on differentiable mani-folds. We shall at first use the same symbol E also for the associated sheaf of C∞

sections, even though later on in this chapter we shall need to resort to a moreprecise notation.

Let X be a differentiable manifold. We shall denote by ΩkX the sheaf of differ-ential k-forms on X. If E is a smooth complex vector bundle on X, a connection∇ on E is a C-linear sheaf morphism

∇ : E → Ω1X ⊗ E

satisfying the Leibniz rule

∇(fσ) = f∇(σ) + df ⊗ σ

for every section σ of E and every function f on X. (The tensor product ofC∞ vector bundles is taken over the sheaf of complex-valued smooth functions.)For every k ≥ 1, the connection ∇ yields in a natural way C-linear morphisms∇ : ΩkX ⊗ E → Ωk+1

X ⊗ E satisfying the Leibniz rule

∇(ω ⊗ σ) = dω ⊗ σ + (−1)kω ∧∇(σ) .

The morphismF∇ : E → Ω2

X ⊗ E , F∇ = ∇ ∇ ,

called the curvature of ∇, is C∞X -linear and therefore may be regarded as a globalsection of the sheaf Ω2

X ⊗End(E). If the curvature F∇ vanishes, the connection issaid to be flat.

5.1. Basic notions 149

We may also consider connections on principal bundles. We recall that aprincipal bundle over a differentiable manifold X with structure group a Lie groupG is a manifold P carrying a free action of G such that the quotient P/G isisomorphic to X, and the projection π : P → X is locally trivial. For every u ∈ P ,the vertical tangent space VuP is defined as ker(π∗ : TuP → Tπ(u)X).

Definition 5.1. A connection Γ on a principal G-bundle P is a smooth G-invariantdistribution HuP ⊂ TuPu∈P such that for all u ∈ P one has TuP = VuP⊕HuP .

4

To the connection Γ we may associate a g-valued 1-form ωΓ on P , where g isthe Lie algebra of G. The 1-form ωΓ is defined as the annihilator of the distributionΓ. The curvature of Γ is the g-valued differential 2-form on P defined by

RΓ = dωΓ + 12 [ωΓ, ωΓ] .

The 2-form RΓ is horizontal (i.e., RΓ(α, β) = 0 if α ∈ VuP and β ∈ TuP ) andG-equivariant, i.e., R∗gRΓ = Adg−1RΓ for all g ∈ G; here Rg is the right action ofthe group, Rg(u) = ug).

If H is a closed subgroup of G, a principal H-bundle Q with an H-equivariantfiber-preserving inclusion j : Q→ P is said to be a reduction of the structure groupG to H. If a connection Γ on P induces a connection on some Q for some closedproper subgroup H, we say that Γ is reducible. If Γ is not reducible, we say it isirreducible.

In the vector bundle case a connection ∇ on a vector bundle E is said to bereducible if there is a direct sum decomposition E = E′ ⊕ E′′ such that ∇(E′) ⊂Ω1X ⊗ E′.

If F is a space over which the group G acts via a representation ρ, and P isa principal G-bundle, we may form the associated bundle E(P, F ) = P ×G F asthe quotient of P × F under the equivalence relation (u, v) ∼ (ug, ρ(g−1)v). If Fis a linear space and ρ is a linear representation, E = E(P, F ) is a vector bundle.In this case, a connection on P induces a connection on E in the former sense. See[185] for details. Note that in particular if G = Gl(n,C) the datum of a principalGl(r,C)-bundle is equivalent to that of a rank r complex vector bundle E.Example 5.2. An important example of associated bundle is the adjoint bundleAdP . This is the bundle with standard fiber g associated to P via the adjointrepresentation of G on g. We shall denote by Ωk(AdP ) the space of differentialk-forms on X with values in AdP . Note that the curvature RΓ of a connection onP may be regarded as an element in Ω2(AdP ), and analogously, the difference oftwo connections in an element in Ω1(AdP ).

The adjoint bundle AdP enters the Atiyah sequence

0→ AdP → TP/G→ TX → 0


and one may regard a connection on P as a splitting of this sequence, cf. [14]. 4

A vertical automorphism of a principal bundle P is a G-equivariant verticaldiffeomorphism φ : P → P (i.e., it verifies π φ = π and φ(ug) = φ(u)g).

The group of smooth (vertical) automorphisms of P , which we denote byG, is usually called the gauge group. It acts naturally on a connection Γ andon the curvature RΓ by pullback. An analogous situation prevails in the vectorbundle case; so, if φ ∈ G, we shall denote by φ∗(∇) and φ∗(F∇) the transformedconnection and curvature, respectively. If U ⊂ X is an open subset over which E

trivializes, after fixing a trivialization on U the curvature F∇ may be regarded asa matrix-valued 2-form, while the restriction of φ to U is described by a smoothmap g : U → Gl(r,C), where r is the rank of E. The transformed curvature maybe written on U as

φ∗(F∇) = Adg−1(F∇) = g−1 F∇ g .

Let as assume now that X is oriented and is equipped with a Riemannianmetric γ. This allows one to introduce the Hodge duality operator ∗ as the map∗ : ΩkX → Ωn−kX (where n = dimX) defined on a global k-form α by the condition

α ∧ ∗β = (α, β) vol(γ)

for all k-forms β (in the right-hand side ( , ) is the scalar product given by theRiemannian metric, and vol(γ) is the Riemannian volume form). Note that ∗2 =(−1)k(n−k) .

5.1.2 Instantons

We can now define the notion of instanton.

Definition 5.3. Let X be an orientable 4-manifold, equipped with a Riemannianmetric, and let E be a vector bundle on X. An instanton on E is a connection ∇whose curvature F∇ is anti-self-dual with respect to Hodge duality, ∗F∇ = −F∇.(This makes sense since F∇ is a 2-form with values in End(E).) Analogously, aninstanton on a principal G-bundle P on X is a connection Γ on P whose curvatureRΓ is anti-self-dual, that is, ∗RΓ = −RΓ (here one regards RΓ as an element inΩ2(AdP ), so that it makes sense to apply the Hodge duality operator to it). 4

Remark 5.4. We might as well define instantons as connections with self-dualcurvature. Since a reversal of the orientation swaps self-dual with anti-self-dual2-forms, as far as there is no distinguished orientation the two notions are in-terchangeable. When X is a complex manifold, so that a preferred orientationdoes exist, and E is a vector bundle, it is preferable to choose the anti-self-dualcondition since in that case instantons relate to stable bundles via the so-calledHitchin-Kobayashi correspondence, see Section 5.1.3. 4


We briefly recall the construction of moduli spaces of instantons, i.e., of aspace which parameterizes gauge equivalence classes of instantons. For technicalreasons we assume that the structure group G is compact and semisimple. Thefirst step is to consider the space of all connections on the bundle P . This is anaffine infinite-dimensional space A, modeled on the vector space Ω1(AdP ). Byintroducing suitable Sobolev norms it can be given the structure of Hilbert mani-fold, cf. [109, 102]. To avoid pathological behaviors, we restrict A to its subspaceA] whose elements are irrreducible connections.

The next step is to take quotient by the natural action of the gauge group G.Provided that one takes completions with respect to suitable norms, the quotientB = A]/G has a structure of smooth, infinite-dimensional Hilbert manifold; it iscalled the orbit space. (For details on the analytical aspects of this constructionthe reader may refer to [109] and [102].) Its points represent gauge equivalenceclasses of irreducible connections on P .

The space of gauge equivalence classes of irreducible anti-self-dual connec-tions M is a subset of B. It is a remarkable fact that M is a finite-dimensional,smooth differentiable manifold. If we want to make a precise statement we needto make assumptions of some sort on the base manifold X. One possibility is toassume that the (oriented and compact) Riemannian 4-manifold (X, γ) is anti-self-dual, i.e., its Weyl curvature 2-form is anti-self-dual and the scalar curvatureof (X, γ) is nonnegative (this is the case discussed in [17]); the Weyl curvature2-form is an invariant of the conformal structure of (X, γ) which is constructedfrom the Riemannian curvature, see, e.g., [40]. Under all these assumptions, onecan prove that M is smooth of dimension

dimM = p1(AdP )− (1− b1 + b+) dimG

where p1(AdP ) is the first Pontrjagin class of the adjoint bundle AdP , b1 =dimH1(X,R) is the first Betti number of X, and b+ is the number of positiveeigenvalues of the intersection form (the quadratic form defined on H2(X,R) bythe cup product). Equivalently, b+ is the dimension of the space of harmonic self-dual 2-forms on X.

The moduli spaceM has a natural metric, called the Weil-Petersson metric.Since the structure group G is semisimple, the Killing-Cartan form κ on the Liealgebra g of G is nondegenerate. One starts by equipping the affine space A with ametric by noticing that the tangent space T∇A at a point ∇ ∈ A may be identifiedwith the space Ω1(AdP ). The Weil-Petersson metric Φ∇ is defined as

Φ∇(α1, α2) =∫X

κ(α1, α2) vol(γ)

where κ(α1, α2) is obtained by first applying the Killing-Cartan form to α1 and α2,thus getting a 2-form on X, and then making it into a function using the metric


γ on X. Moreover, vol(γ) is the volume form on X given by the metric γ. Themetric Φ∇ is gauge-invariant, hence descends to the orbit space B, and induces ametric on M by restriction.

The product manifold X×B carries a universal G-bundle Q equipped with auniversal connection ∇∇∇: if b ∈ B is a point corresponding to the gauge equivalenceclass of a connection ∇ on a principal G-bundle P on X, then Q|X×b ' P

and the connection ∇∇∇|X×b is gauge-equivalent to ∇. A very neat and concisetreatment of the main properties of the universal pair (Q,∇∇∇) is given in [19]. Letus here briefly summarize the construction of the pair (Q,∇∇∇).

We consider the action of the gauge group G on the product P × A]. Thisaction has no fixed points so that one can define a principal bundle Q′

Q =P ×A]

G.

The action of the structure group G of P commutes with the action of the gaugegroup, so that G acts on the bundle Q. Since we are considering only irreducibleconnections, this action is free and defines a principal bundle with total space Qand base manifold Q/G ' X × B.

The connection ∇∇∇ is defined as follows. Consider on the space X × A] themetric given by the metric γ on X and the Weil-Petersson metric on A]. This isinvariant under the natural action of G × G and hence descends to a G-invariantmetric on Q. The connection is obtained by considering the distribution in TQwhich is orthogonal to the fibers of Q.

The universality of (Q,∇∇∇) means the following. Let R be a principal G-bundle R on X × Y , where Y is any compact manifold, with the property thatRX×y ' P for all y ∈ Y , and let ∇ be a connection on R. Then there exista map f : Y → B and a principal bundle map f : R → (IdX × f)∗Q such thatf∗(∇∇∇) = ∇.

The curvature F of the universal connection∇∇∇ may be described explicitly. Itis convenient to split it into its Kunneth components with respect to the productX × B, namely, F = F2,0 + F1,1 + F0,2. One has:

1. if b ∈ B, then F2,0|X×b = F∇, where ∇ is a connection whose gauge equiva-

lence class is b;

2. if (v, α) ∈ T(x,b)(X × B) then F1,1(v, α) = α(v), after identifying α with anelement in Ω1(AdP ) such that ∇∗(α) = 0, where ∇ is a connection whosegauge equivalence class is b (one should note that TbB ' Ω1(AdP )/ ker∇∗);

3. if α, β ∈ TbB, then F0,2(α.β) = G∇(λα(β)), where G∇ is the Green functionof the trace (or “rough”) Laplacian ∇∗ ∇ : Ω0(AdP ) → Ω0(AdP ) (i.e.,


G∇ = (∇∗ ∇)−1), while the map λα : Ω0(AdP )→ Ω1(AdP ) is defined byλα(s) = [α, s].

Let us eventually notice that if we choose a representation ρ : G → Aut(V )of G, then we may define on X ×B a universal bundle QV with standard fiber Vand a universal connection.

The bundles Q, QV and the corresponding connections may be restricted toX ×M ⊂ X × B, obtaining universal bundles with connections for the instantonmoduli spaces.

5.1.3 The Hitchin-Kobayashi correspondence

In some cases, as we saw in Chapters 3 and 4, the Fourier-Mukai transform pre-serves the stability of the sheaves it acts on. A parallel property of the Nahmtransform is that sometimes it maps instantons to instantons. As we shall see inthis chapter, the two transforms may be related, and the above-mentioned preser-vation properties are intertwined by the correspondence between instantons, ormore general, Hermitian-Yang-Mills bundles, and stable bundles — the so-calledHitchin-Kobayashi correspondence, which we proceed now to discuss briefly. Com-prehensive references on this subject are the monographs [184, 200].

Let X be an n-dimensional compact Kahler manifold, with Kahler form ω.One can give a notion of µ-stability for coherent sheaves F on X exactly as in theprojective case by defining the degree of F as

deg(F) =∫X

γ1(F) ∧ ωn−1

where γ1(F) is here any closed 2-form on X whose cohomology class is c1(F) (oneshould notice that c1(F) can be introduced for every coherent sheaf F even whenX is not projective by defining it as the first Chern class of the determinant bundledet(F)).

On the other hand, let E be a holomorphic vector bundle on X equippedwith a Hermitian fiber metric h. The latter, together with the complex structureof E, singles out a unique connection ∇ on E, which has the property of beingcompatible with both the metric h, meaning that

dh(s, t) = h(∇(s), t) + h(s,∇(t))

for all sections s, t of E, and with the complex structure of E, which in turnmeans that ∇0,1 = ∂E , where ∂E is the Dolbeault (Cauchy-Riemann) operator ofthe bundle E. This connection is called the Chern connection of the Hermitianbundle (E, h) [184].


Let Λ be the adjoint of the map given by wedging by the Kahler form, i.e.,(Λ(α), β) = (α, ω ∧ β) for all forms α, β on X.

Definition 5.5. A Hermitian vector bundle (E, h) is said to satisfy the Hermitian-Yang-Mills condition if there exists a complex constant c such that

Λ(F∇) = c IdE

where F∇ is the curvature of the Chern connection ∇. 4

Remark 5.6. The constant c is fixed by the topology of the bundle, and one hasindeed

c =2nπ

n! vol(X)µ(E) with vol(X) =

1n!

∫X

ωn .

4Remark 5.7. The notion of Hermitian-Yang-Mills bundle generalizes that of in-stanton: indeed, if n = 2 one may see that Hermitian-Yang-Mills bundles of zerodegree are exactly the instantons. 4

Definition 5.8. A coherent sheaf F is said to be polystable if it is a direct sum ofµ-stable sheaves having the same slope. 4

Proposition 5.9. The sheaf of holomorphic sections of a Hermitian-Yang-Millsbundle is polystable.

The proof of this result is not difficult, and may be found, e.g., in Kobayashi[184]. A much deeper result is the converse.

Theorem 5.10. A µ-stable bundle E on a compact Kahler manifold admits a Hermi-tian metric h (unique up to homotheties) such that (E, h) satisfies the Hermitian-Yang-Mills condition.

The proof given by Donaldson first for projective surfaces, and then for pro-jective varieties of any dimension [99, 100], considers the space of all Hermitianstructures on E and defines a parabolic flow on it by introducing a suitable func-tional. The proof that the flow admits a limit, which is the sought-for Hermitian-Yang-Mills metric, relies on the Mehta-Ramanathan theorem about the restrictionof semistable sheaves to divisors in certain linear systems, and therefore confinesthe validity of the proof to the projective case. This techniques is nicely illustratedin Kobayashi [184]. A proof which works on general compact Kahler manifolds waslater given by Uhlenbeck and Yau [286].

Example 5.11. As an application of the Hitchin-Kobayashi correspondence we mayshow how Proposition 4.74 yields a statement on some moduli spaces of instantons.Let X be a K3 surface, and let H, ` be divisors that make it strongly reflexive (see


Chapter 4). We saw in Proposition 4.74 that the moduli space Mn of H-stablesheaves on X with Chern character (1+2n,−n`,−5n) is connected and projectiveand that it contains no µ-stable sheaves. Identifying µ-stable bundles of zero degreewith irreducible instantons, this means that on X there are no irreducible U(2n+1)-instantons, with fixed determinant OX(−n`) and second Chern character −5n,and the moduli space of all instantons with this type is isomorphic to the n-thsymmetric product SnX. This last fact follows from the structure of the so-calledUhlenbeck compactification of the instanton moduli space, see, e.g., [102]. 4

5.1.4 Dirac operators and index bundles

Finally, we review another basic construction needed to define the Nahm trans-form, namely, the Dirac operator. A suitable reference on this topic, among manyothers, is Lawson and Michelsohn’s book [194]. If X is an orientable Riemannianmanifold, with Riemannian metric γ, we may consider the principal bundle SO(X)of oriented orthonormal frames (bases of tangent spaces at the points of X), whosestructure group is the special orthogonal group SO(n), where n = dimX. A spinstructure on X is a principal Spin(n)-bundle Spin(X) on X together with a bun-dle homomorphism Spin(X) → SO(X) which on the fibers reduces to the spincovering homomorphism Spin(n) → SO(n). The second Stieffel-Whitney classw2(X) ∈ H2(X,Z2) is an obstruction to the existence of a spin structure on X.When w2(X) = 0 the isomorphism classes of spin structures on X are classifiedby the group H1(X,Z2).

Let Cl(X, γ) be the Clifford algebra bundle of (X, γ), that is, the bundle ofClifford algebras on X whose fiber at the point x is the Clifford algebra associatedwith the vector space TxX equipped with the quadratic form given by the scalarproduct γ(x). A Clifford bundle S on X is a bundle of Cl(X, γ)-modules equippedwith a Hermitian metric and a connection ∇ satisfying suitable and natural com-patibility conditions, see [194]. The Dirac operator is the first-order differentialoperator D : Γ(S)→ Γ(S) defined by the composition of the maps

Γ(S) ∇−→ Γ(Ω1X ⊗ S)

γ−1

−−→ Γ(TX ⊗ S) Cl−−→ Γ(S) .

Here Cl : Γ(TX⊗S)→ Γ(S) is the multiplication morphism given by the Cl(X, γ)-module structure of S and the immersion TX → Cl(X, γ).

An important example of Clifford bundle S is provided by the spinor bundle,i.e., the bundle associated with the principal bundle Spin(X) via the spin rep-resentation of the complexified Clifford algebra Cl(Rn, γ0) ⊗ C, where γ0 is thestandard scalar product in Rn. The bundle S has complex rank 2[n/2], and as aconsequence of the Z2-gradation of the Clifford algebra it inherits a Z2-gradationas well, S = S+ ⊕ S− (using a terminology coming from physics, we may call S±


the bundles of spinors of positive or negative chirality). The Dirac operator hasodd degree with respect to this gradation. By abuse of notation, usually one writesD for the operator Γ(S+)→ Γ(S−); the adjoint D∗ then coincides with the termof the full Dirac operator mapping Γ(S−) to Γ(S+).

In order to apply standard techniques in analysis one needs to completethe space Γ(S) to a Hilbert space. Assuming that X is compact and using theRiemannian metric on X and the Hermitian fiber metric in S, we may consider theL2 norm on the space Γ(S), and then complete the latter in this norm, obtainingthe Hilbert space L2(S). More generally, for every integer p ≥ 0 we may considerthe Hilbert space L2

p(S) — called the Sobolev space of sections of S of weight p —formed by those sections of S whose p-th covariant derivative has finite L2 norm.The Dirac operator extends to an operator L2

1(S)→ L2(S), or more generally, toan operator L2

p(S)→ L2p−1(S) for all integers p ≥ 1. Elliptic regularity implies that

the kernel of any of these operators coincides with the kernel of D : Γ(S)→ Γ(S).Moreover, due to the fact that D is a Fredholm operator, this kernel is finite-dimensional. The same is true for the cokernel of this operator, so that it makessense to introduce the index of the Dirac operator as the integer number

ind(D) = dim ker(D)− dim coker(D) .

This number is actually a topological invariant of the manifold X, and is computedby the celebrated Atiyah-Singer index theorem (we assume henceforth the thedimension of X is even):

ind(D) =∫X

A(X) (5.1)

where A(X) is a characteristic class that may be expressed in terms of the Pon-trjagin classes of X. A simple way of writing a formula for A(X) (in de Rhamcohomology) is

A(X) =[p

(i

2πR

)]where R is the curvature of a connection on the tangent bundle to X, and p is thepolynomial which expresses the formal Taylor expansion of the function

f(z) =12

√z

sinh 12

√z

around z = 0 up to order n/2. A nontrivial consequence of the index formula (5.1)is that the right-hand side (called the A-genus of X) is an integer.

There exists a twisted version of Atiyah-Singer index theorem: assume that avector bundle E is given, with a connection ∇ on it, and define the twisted Diracoperator

D∇ : Γ(E ⊗ S+)→ Γ(E ⊗ S−),

D∇(s⊗ ψ) = ∇(s) · ψ + s⊗D(ψ)


where again the product · is obtained by first applying the inverse Riemannianmetric to ∇(s) and then performing the Clifford product. The formula (5.1) shouldnow be replaced by

ind(D∇) =∫X

ch(E) A(X) .

A far reaching generalization of these formulas is the Atiyah-Singer indextheorem for families. In this case we deal with a family of Dirac operators, whichwe may regard as a continuous map D : T → H, where T is a topological space(playing the role of parameter space) and H is the (separable) Hilbert space ofbounded operators L2

1(E ⊗ S+) → L2(E ⊗ S−). Let us denote Dt = D(t), i.e.,Dt is the Dirac operator corresponding to the parameter t ∈ T . By assigning toeach point t ∈ T the virtual vector space ker(Dt) − coker(Dt) one constructs aclass ind(D) in the topological K-theory group K(T ) of the parameter space T .The Atiyah-Singer theorem for families is a formula which computes the Cherncharacter of the virtual bundle ind(D). We shall give here that formula only ina specific but very important case. Let W be a vector bundle on the productX × T , equipped with a connection ∇∇∇. Assume that the restriction W|X×t isisomorphic to E for every t ∈ T . Then the connections∇t =∇∇∇|X×t yield a familyof connections on E, and we then twist the Dirac operator with that family. Inthat case one has

ch(ind(D)) =∫X

ch(W) A(X) ; (5.2)

here the integral∫X

may be regarded as the integration along the fibers of theprojection X → Y → Y .

One can define a generalization of the spin structures, and introduce a re-lated Atiyah-Singer index theorem, by using complex geometry. Let X be an n-dimensional complex manifold with a Hermitian metric h, and define the complexvector bundle

S = ⊕nk=0Ω0,kX

with the obvious Z2-gradation. This becomes a Clifford module by letting, forevery vector field v on X and every section η of S

v · η =√

2(h(p) ∧ η − iqη)

where p and q are the (1,0) and (0,1) parts of v, respectively. One can prove thatif the pair (X,h) is a Kahler manifold, then the Dirac operator of this Cliffordmodule may be identified with the operator

√2(∂ + ∂∗)

where ∂ is the Dolbeault (Cauchy-Riemann) operator. If we twist this operatorwith a complex holomorphic vector bundle E we obtain a twisted Dirac operator


DE , and one can easily show that the index of this operator coincides with theholomorphic Euler characteristic of E, namely,

ind(DE) = χ(E) =n∑i=0

(−1)i dimHi(X,E) .

The Atiyah-Singer index theorem (5.2) in this case takes the form

χ(E) =∫X

ch(E) td(X) ,

where td(X) is the Todd class of X, i.e., it reproduces the Hirzebruch-Riemann-Roch theorem for a holomorphic vector bundle (we are of course assuming thatX is compact). In the same way, if X is a compact Kahler manifold, Y is acomplex manifold, and E is a holomorphic complex vector bundle on X × Y , theAtiyah-Singer index theorem for families reproduces the Grothendieck-Riemann-Roch theorem for the Chern character of the higher direct images of E:

n∑i=0

(−1)i ch(Riπ∗E) =∫X

ch(E) td(X)

where π : X × Y → Y is the projection.

5.2 The Nahm transform for instantons

5.2.1 Definition of the Nahm transform

Let (X, γ) be a smooth oriented Riemannian spin 4-manifold such that the corre-sponding scalar curvature Rγ is nonnegative at every point of X. For simplicity,we assume that X is compact. We denote as before by S± the spinor bundles ofpositive and negative chirality. Moreover, let Y be a smooth manifold parame-terizing a family of anti-self-dual connections on a fixed complex vector bundleW → X. We assume that the bundle W is equipped with a Hermitian metric, andthat the connections we are considering on it are compatible with this metric (sothat they are unitary connections). Associating to any connection in this family itsgauge equivalence class, we obtain a map f : Y →M, whereM is a moduli spaceof instantons on W . We assume that the map f is smooth. We also assume thatY has a Riemannian metric compatible via the map f with the Weil-Peterssonmetric on M. So for every t ∈ Y one has an anti-self-dual connection ∇t on thebundle W .

The Nahm transform from X to Y is a mechanism that transforms Hermi-tian vector bundles with unitary anti-self-dual connections on X into Hermitian

5.2. The Nahm transform for instantons 159

vector bundles with unitary connections on Y . If Y parameterizes a family of flatconnections over X, we will say that the transform is flat; otherwise, we will saythat the Nahm transform is nonflat.

Let us now describe the transform in detail. Let E be a Hermitian com-plex vector bundle on X with an anti-self-dual connection ∇ compatible with theHermitian metric (i.e., we have a unitary anti-self-dual connection). On the ten-sor bundle E ⊗W → X, we have a twisted family of anti-self-dual connections∇t = ∇⊗ IdW + IdE ⊗∇t. We further assume an irreducibility condition.

Definition 5.12. The family ∇t is 1-irreducible if for all t ∈ Y the condition∇ts = 0 implies s = 0. 4

In other terms, for all values of t, the tensor bundle E⊗W has no covariantlyconstant sections with respect to the connection ∇t. We consider the family D oftwisted Dirac operators

Dt: L2

p(E ⊗W ⊗ S+)→ L2p−1(E ⊗W ⊗ S−),

and denote as usual by let D∗t the adjoint Dirac operator. The Dirac LaplacianD∗tDt is related to the trace Laplacian ∇lt ∗ ∇t via the Weitzenbock formula:

D∗tDt = ∇∗t ∇t − F+t + 1

4Rγ (5.3)

= ∇∗t ∇t + 14Rγ (5.4)

since the self-dual part F+t of the curvature Ft of ∇t vanishes. Applying (5.3) to

a section s ∈ L2p(E ⊗W ⊗ S+) and integrating by parts we obtain

||Dts||2 = ||∇ts||2 + 14

∫X

Rγ〈s, s〉 ≥ 0 (5.5)

with equality if and only if s = 0, since Rγ ≥ 0.

Therefore, we conclude that kerDt = 0 for all t ∈ Y . This means thatE = −ind(D) is a well-defined complex vector bundle over Y ; the fiber Et is givenby coker(Dt). Let H± denote the trivial Hilbert bundle over Y with fibers givenby the spaces L2

p−1(E ⊗W ⊗ S±). One can think of E as a subbundle of H−; wehave an exact sequence of bundles

0→ Eι−→ H− → H+ → 0

which may be split using the metric on the bundle H−, thus defining a projectionP : H− → E. This provides E with a natural Hermitian metric, and we can alsodefine a unitary connection ∇ on E via the projection formula

∇ = P dH− ι (5.6)


where dH− denotes the trivial covariant derivative on H−, i.e., the exterior differ-ential.

It is easily checked that this construction behaves well with respect to gaugetransformations of W . This means that if we apply an automorphism of the bundleW , thus getting a new family ∇′t of connections on W , and a new family of Diracoperators D′, there is a natural isomorphism coker(D) ' coker(D′), so that theindex bundle E descends to a bundle on the quotient Y/G, where G denotes thegroup of gauge transformations of W (we are assuming here that this quotientis well behaved). Moreover, a connection ∇ is defined on this bundle. For thisreason, we may assume that Y parameterizes a family of gauge equivalence classesof anti-self-dual connections on the fixed vector bundle W sucht that the familyof connections twisted by ∇ is 1-irreducible.

Definition 5.13. The pair (E, A) is called the Nahm transform of (E,A). 4

The Nahm transform is well behaved also with respect to the gauge trans-formations of E.

Lemma 5.14. If ∇ and ∇′ are two gauge-equivalent connections on the vector bun-dle E → X, then ∇ and ∇′ are gauge equivalent connections on the transformedbundle E → Y .

Proof. Let h be a a bundle automorphism h : E → E which makes ∇ and ∇′gauge equivalent and let g = h ⊗ IdW be the induced automorphism of E ⊗W .Then D∗t′ = g−1D∗Atg, for all t ∈ Y . Thus if Ψi is a basis for kerD∗t , thenΨ′i = g−1Ψi is a basis for kerD∗t′ . So g can also be regarded as an automorphismof the transformed bundle E. It is then easy to see that

∇′ = P ′ dH− ι′ =(g−1P ′g

) dH−

(g−1ι′g

)= g−1∇g

since dH− g−1 = 0, for g does not depend on t.

The construction performed in this section provides the following result.

Theorem 5.15. Let Y be a connected component of the moduli space of irreducibleinstanton connections on a smooth oriented Riemannian spin 4-manifold X withnonnegative scalar curvature, and let E be a Hermitian complex vector bundle onX. Let W be a Hermitian complex vector bundle on X carrying a 1-irreduciblefamily of anti-self-dual connections. The corresponding Nahm transform yields awell-defined map from M(E), the moduli space of gauge equivalence classes ofunitary anti-self-dual connections on E, into the space BY (E) gauge equivalenceclasses of (unitary) connections on the Nahm transform E.


5.2.2 The topology of the transformed bundle

The Atiyah-Singer index theorem for families allows us to compute the Cherncharacter of the transformed bundle. Let us assume that there exists a complexvector bundle W on X × Y with a connection ∇∇∇, such that for all t ∈ Y one hasW|X×t 'W and ∇∇∇|X×t = ∇t. Then the formula (5.2) yields

ch E = −∫X

ch(E) ch(W) A(X)

where the minus sign is needed because E is the bundle of cokernels. This formulashows that the topology of the transformed bundle E depends only on the topologyof the original bundle E.

Example 5.16. Let us now briefly analyze the Nahm transform for the simplestpossible compact spin 4-manifold with nonnegative scalar curvature, the round4-dimensional sphere S4. So let X = S4, and let Y be the moduli space of SU(2)instantons over S4 with charge one; as a Riemannian manifold, Y is a hyperbolic5-ball B5 [109]. Let E → S4 be a complex vector bundle of rank n ≥ 2, equippedwith an instanton ∇ of charge k ≥ 1. Nahm transform gives a bundle E → B5 ofrank 2k + r, by the index formula (5.2). Since B5 is contractible, this is the onlynontrivial topological invariant of the transformed bundle. 4Remark 5.17. (Differential properties of the transformed connection.) Given thatthe original connection ∇ satisfies a nonlinear first-order differential equation (theanti-self-duality condition), it is reasonable to expect that the transformed con-nection will also satisfy some kind of strict differential or algebraic condition.However, since it does not seem possible to write a formula for the curvature ofthe transformed connection ∇ which depends explicitly on the curvature of theoriginal connection ∇, it is in general very difficult to characterize any particularproperties of ∇.

For instance, when the parameter space Y is 4-dimensional, one would liketo know whether F∇ is anti-self-dual. This seems to be a very hard question ingeneral. However, when M is a hyperkahler manifold, complex analytic methodscan be used to show that this is indeed the case. This will be shown later on ina more general setting. The results we shall discuss will include as a special casethe original Nahm transform on flat tori, which is known to map instantons on a4-torus to instantons on the dual torus [57, 102, 263]. This case will be discussedin Section 5.2.4, 4

5.2.3 Line bundles on complex tori

As a preparation for studying the Nahm transform on tori, we analyze in thissection a very handy description of U(1) line bundles on complex tori in terms of


their automorphy factors. A standard reference about this theory is [42].

Let V be a g-dimensional complex vector space, and Ξ a nondegenerate latticein it. Then the quotient T = V/Ξ has a natural structure of g-dimensional complexmanifold, and is said to be a complex torus of dimension g. If some conditions aresatisfied (the Riemann bilinear conditions), T is actually algebraic, and then it isan Abelian variety, whose origin is the image of the origin of V in T . However inthis section we consider the general case where T may not be algebraic.

Since any generator of the lattice Ξ corresponds to a loop in T , we have anatural identification of Ξ with the fundamental group π1(T ), and hence with thehomology group H1(T,Z). As a result, we also have identifications Hk(T,Z) 'ΛkΞ∗.

Let H(T ) be the space of Hermitian forms H : V × V → C that satisfy thecondition Im(H(Ξ,Ξ)) ⊂ Z. Under the inclusion H2(T,Z) ⊂ Hom(V × V,C) wehave an identification of H(T ) with the image of the morphism

c1 : Pic(T )→ H2(T,Z) ,

i.e., H(T ) coincides with the Neron-Severi group NS(T ).

Definition 5.18. A semicharacter associated with an element H ∈ H(T ) is a mapχ : Ξ→ U(1) such that

χ(λ+ µ) = χ(λ)χ(µ) eiH(λ,µ) .

An element H ∈ H(T ) and an associated semicharacter χ define a automor-phy factor

a : V × Ξ → U(1)

a(v, λ) = χ(λ) eπH(v,λ)+π2H(λ,λ) .

4

If H = 0, then an associated semicharacter is just a character of the latticeΞ.

In the same way as functions on T are just periodic functions on the uni-versal cover V , sections of a line bundle L on T are functions on V that satisfya “generalized periodicity condition” given by an automorphy factor. We haveindeed:

Proposition 5.19. The holomorphic functions s on V that satisfy the condition

s(v + λ) = a(v, λ) s(v)

for all v ∈ V and λ ∈ Ξ, where a is an automorphy factor associated with anelement H ∈ H(T ), are in a one-to-one correspondence with sections of a linebundle L on T such that c1(L) = H.


For a proof see [42].

In this framework the dual torus T ∗ may be built as follows. Let Ω be theconjugate dual space of V , and let

Ξ∗ = ` ∈ Ω | `(Ξ) ⊂ Z

be the lattice dual to Ξ. Then we set T ∗ = Ω/Ξ∗. One has a natural isomorphismT ∗ ' HomZ(Ξ, U(1)), and an exact sequence

0→ T ∗ → Pic(T )→ NS(T )→ 0

which shows that the dual torus T ∗ parameterizes flat U(1) bundles on T (andindeed when T is algebraic, T ∗ is the complex torus underlying the dual Abelianvariety T ). If ξ ∈ T ∗, the line bundle on T parameterized by ξ may be describedby the automorphy factor

aξ(v, λ) = eπ w(λ) (5.7)

where w is a representative of ξ in Ω.

On the basis of Proposition 5.19 we can give a very explicit description ofthe Poincare bundle P on T × T ∗. We choose the element H ∈ H(T × T ∗) givenby

H(v, w, α, β) = β(v) + α(w) (5.8)

where v, w ∈ V , α, β ∈ Ω, and an associated semicharacter

χ(λ, µ) = eiπ µ(λ) . (5.9)

Note that H2(T × T ∗,Z) contains the Kunneth component H1(T,Z)⊗H1(T ∗,Z)which can be identified with EndZ(H1(T,Z)), and under this identification H

corresponds to the identity endomorphism.

The Poincare bundle is by definition the line bundle P corresponding to theHermitian form (5.8) and semicharacter (5.9). Thus, its automorphy factor is

aP(v, w, λ, µ) = eiπ µ(λ) eπ(µ(v)+w(λ)) .

Let us check that that P|T×ξ is isomorphic to the flat line bundle on T corre-sponding to ξ ∈ T ∗. Indeed P|T×ξ admits an automorphy factor given by

aP(v, w, λ, 0) = eπ w(λ) .

By comparing with Definition 5.18 and Equation 5.7 we see that P|T×ξ is iso-morphic to the line bundle parameterized by ξ.


5.2.4 Nahm transform on flat 4-tori

Let us describe here the special case of the general construction of the Nahmtransform described in Section 5.2.1 that one obtains when the manifold X, Y area flat 4-tours and its dual. This is the original transform defined by Nahm [230]and later formalized by Braam and van Baal [57], Schenk [263] and Donaldsonand Kronheimer [102]. Let us equip X with a compatible complex structure (inother terms, since X is a quotient V/Ξ, where V is 4-dimensional vector spaceand Ξ ∈ V and nondegenerate lattice, we equip V with a complex structure), andidentify Y with the dual complex torus. So Y parameterizes flat U(1) line bundleson X, and we have the Poincare bundle P on X × Y . For every ξ ∈ Y the linebundle Pξ is trivial as a C∞ bundle, so that we may regard the collection Pxia trivial bundle W with a varying complex structure and Hermitian metric.

Let E be a holomorphic Hermitian bundle on X with a compatible anti-self-dual connection ∇. Donaldson and Kronheimer considered the following irre-ducibility condition.

Definition 5.20. The pair (E,∇) is said to be without flat factor if there is no∇-compatible splitting E = E′ ⊕ L where L is a flat line bundle. 4

As a matter of fact this is just 1-irreducibility in disguise:

Proposition 5.21. (E,∇) is without flat factors if and only if the family of connec-tions ∇t = ∇⊗ Id+Id⊗∇ξ (where ∇ξ is the connection on Pξ) is 1-irreducible.

Proof. Assume that ∇t is not 1-irreducible. Then for some ξ ∈ Y there is asection s of E ⊗Pξ such that ∇ξ 6= 0. After tensoring by P−ξ, this splits off a flatparallel summand of E (i.e., E = E′⊕P−ξ) and contradicts the fact the (E,∇) iswithout flat factors.

Conversely, if E has a parallel splitting E = E′⊕L with L flat, then L ' Pξfor some ξ ∈ Y . Let s be a nonzero covariantly constant section of L, i.e., ∇ξ(s) =0. Then s, regarded as a section of E, satisfies ∇ξ(s) = 0.

So, if the pair (E,∇) is without flat factors, we may apply the general theoryof Section 5.2.1 and obtain a holomorphic Hermitian bundle E on Y equippedwith a compatible connection ∇. One can prove that ∇ is anti-self-dual. Since thisis particular case of the general result proved in Section 5.4.3, we shall not repeatit here.

Moreover, in this case the Nahm transform is invertible, its inverse beingexactly the same transform when we identify the dual torus to Y as X, and thecorresponding Poincare bundle on Y × X as P∗. The proof that this actuallyprovides an inverse to the Nahm transform from X to Y may be given in terms ofa direct computation, as in [57] (which follow closely [230]). When X is algebraic,

5.3. Compatibility between Nahm and Fourier-Mukai 165

we can use the identification of the Nahm transform with the Abelian Fourier-Mukai transform which follows from Section 5.3, and then use the fact that, asshown in Chapter 3, the Abelian Fourier-Mukai transform, being an equivalence ofcategories, is invertible (in Chapter 3 we consider Abelian varieties, but the proofalso works for complex tori). This is very much in the spirit of the proof given byDonaldson and Kronheimer [102].

5.3 Compatibility between Nahm and Fourier-Mukai

Given a submersive morphism of complex manifolds f : Z → Y , and a complexvector bundle E on Z, there is a relationship between the higher direct images ofE (the sheaf of holomorphic sections of E) and the index of the relative Dolbeaultcomplex twisted by E. In this section we analyze this correspondence. This will beused to study the relationship between the Nahm transform and the Fourier-Mukaifunctors.

5.3.1 Relative differential operators

Let f : Z → Y be a submersion of differentiable manifolds, and let E → Z andF → Z be complex vector bundles. Let us denote by E∞, F∞ the locally freeC∞Z -modules of sections of E and F .

Definition 5.22. A relative differential operator of order k is a morphism

D : f∗E∞ → f∗F∞

of C∞Y -modules that factors through the direct image of the sheaf J k(E∞/Z) ofsections of the k-order (relative) jet bundle Jk(E/Z)→ Z,

f∗E∞D //

f∗F∞

J k(E∞/Z)

99rrrrrrrrrr

4

Definition 5.23. Assume that E and F have Hermitian fiber metrics. The adjointD∗ : f∗F∞ → f∗E∞ of a relative differential operator D : f∗E∞ → f∗F∞ is definedby letting (u,Dv) = (D∗u, v) for each pair of sections u, v, of f∗E∞ and f∗F∞respectively, on an open subset V ⊂ Y . 4

If y ∈ Y is a point in the parameter space Y , we denote Zy = f−1(y)and by Ey the restricted fiber bundle E ×Y y → Zy. One easily checks that


J k(E∞/Z)y = Γ(Zy,J k(Ey∞/Zy)). A relative differential operator D : f∗E∞ →f∗F∞ induces differential operators (in the usual sense)

Dy : Γ(Zy, Ey∞)→ Γ(Zy,Fy∞)

and may be therefore regarded as a family of differential operators parameterizedby points of Y .

Definition 5.24. A relative differential operator D : f∗E∞ → f∗F∞ is elliptic if atany point y ∈ Y the differential operator Dy is elliptic. 4

We recall that a differential operatorD : Γ(U)→ Γ(V ), where U , V are vectorbundles on some differentiable manifold X, is said to be elliptic if for all ξ ∈ andξ ∈ T ∗xX the associated symbol map σx,ξ(D) : Ux → Vx is a linear isomorphism.For details on this notion, the reader may consult, e.g., [54].

If f : Z → Y is proper, then, given an elliptic relative differential operator,the Atiyah-Singer index theory for families provides an element ind(D) ∈ K(Y )called the index of D. If either one of kerDy or cokerDy has constant rank, thenkerD and cokerD are locally free C∞Y -modules of finite rank and one has that

ind(D) = [kerD]− [cokerD] ,

where [ ] denotes a class within K(Y ) (cf. [18], or [158, Appendix II]).

5.3.2 Relative Dolbeault complex

Let Z be a complex manifold and E a holomorphic vector bundle on Z. The sheafof holomorphic sections of E will be denoted by E . In other words, E∞ = E⊗OZ C∞Z .On the other hand, a C∞ vector bundle E → Z whose sheaf of sections is E∞ hasa compatible holomorphic structure if there exists a locally free OZ-submoduleE → E∞ of finite rank that induces an isomorphism E∞ ∼→ E ⊗OZ C∞Z of C∞Z -modules.

We recall the following standard result in the theory of complex vector bun-dles (cf. [184]).

Proposition 5.25. A C∞ Hermitian vector bundle E → Z admits a compatibleholomorphic structure if and only if there exists a Hermitian connection on E → Z

whose curvature is of type (1, 1).

If Ω1Z denotes the sheaf of C∞ 1-forms on Z and ∇ : E∞ → Ω1

Z ⊗C∞Z E∞ is

the aforementioned connection, the sheaf of holomorphic sections of E is given byE = ker ∂E , where ∂E = ∇0,1 : E∞ → Ω0,1

Z ⊗C∞Z E∞ is the (0, 1) component of ∇.


This is actually part of a more general statement, namely, the exactness of theDolbeault sequence of sheaves of OZ-modules:

0→ E → E∞ ∂E→ Ω0,1Z ⊗ E

∞ ∂E→ Ω0,2Z ⊗ E

∞ ∂E→ . . .

Let f : Z → Y be a holomorphic morphism of complex manifolds. Let usdenote by f−1, f∗h , f∗∞ the inverse images in the categories of Abelian sheaves,holomorphic sheaves and C∞ sheaves, respectively. That is, for every OY -moduleE , f∗hE is the OZ-module f∗hE = f−1E ⊗f−1(OY ) OZ , and for every C∞Y -moduleE∞, f∗∞E∞ is the C∞Z -module f∗∞E∞ = f−1E∞⊗f−1C∞Y C

∞Z . In particular, if E∞ =

E ⊗OY C∞Y , then f∗∞E∞ ∼→ f∗hE ⊗OZ C∞Z .

Definition 5.26. The sheaf of relatively holomorphic functions on Z is the sheaff∗hC∞Y . Analogously, we say that a C∞ vector bundle E → Z has a relative holomor-phic structure if there exists a locally free finite-rank f∗hC∞Y -submodule Er → E∞that induces an isomorphism of C∞Z -modules E∞ ∼→ Er ⊗f∗hC∞Y C

∞Z . 4

We shall henceforth assume that the map f : Z → Y is submersive. Then wehave an exact sequence of OZ-modules

0→ f∗∞Ω1Yf∗→ Ω1

Z → Ω1Z/Y → 0 .

The pullback of 1-forms via a holomorphic map preserves the Hodge decomposi-tion, so that we can define a Hodge decomposition for the sheaf of relative 1-forms,Ω1Z/Y = Ω0,1

Z/Y ⊕Ω1,0Z/Y and the corresponding Hodge decomposition for the exterior

powers ΩmZ/Y =∧mΩ1

Z/Y ,

ΩmZ/Y =⊕

p+q=m

Ωp,qZ/Y .

If E → Z is a C∞ vector bundle, and E∞ is the C∞Z -module of its sections,a relative connection for E is a morphism

∇E/Y : E∞ → Ω1Z/Y ⊗C∞Z E

∞

of f−1C∞Y -modules satisfying an obvious Leibniz condition. The (0,1) componentof the relative connection (with respect to the relative Hodge decomposition) is amorphism of f∗hC∞Y -modules

∂E/Y : E∞ → Ω1Z/Y ⊗C∞Z E

∞

which defines a sequence of sheaf morphisms

0→ ker ∂E/Y → E∞∂E/Y→ Ω0,1

Z/Y ⊗ E∞ ∂E/Y→ Ω0,2

Z/Y ⊗ E∞ ∂E/Y→ . . .


The kernel ker ∂E/Y is the sheaf of sections of E that are holomorphic along thefibers of Z → Y . If the (0,2) component of the curvature R = ∇E/Y ∇E/Yvanishes, the sequence (5.3.2) is a complex, called then relative Dolbeault complex.Moreover the equation ∂2

E/Y = 0 is the integrability condition of the equation∂E/Y (s) = 0, so that the relative Dolbeault complex is actually exact, and providesa resolution of ker ∂E/Y by fine sheaves. As a consequence, the higher direct imagesRif∗ ker ∂E/Y are the cohomology sheaves of the complex f∗(Ω

0,•Z/Y ⊗C∞Z E

∞).

A Hermitian relative connection is a relative connection compatible with agiven Hermitian metric in the usual sense. Proposition 5.25 has now a relativeanalogue. Summing up, we have proved the following results.

Proposition 5.27. A Hermitian C∞ vector bundle E → Z admits a compatiblerelative holomorphic structure if and only if there exists a Hermitian relative con-nection on E whose curvature is of type (1, 1).

Proposition 5.28. Let E → Z be a vector bundle endowed with a Hermitian metricand a relative holomorphic structure. The higher direct images Rif∗Er are thecohomology sheaves of the relative Dolbeault complex f∗(Ω

0,•Z/Y ⊗C∞Z E

∞).

Definition 5.29. Let E → Z be a C∞ vector bundle endowed with a relativeholomorphic structure.

1. E satisfies the i-th weak index theorem condition (i.e., it is WITi) ifRjf∗Er =0 for every j 6= i;

2. E satisfies the i-th index theorem condition (i.e., it is ITi) if Hj(Zy, Ey) = 0for every j 6= i and for all points y ∈ Y , where Ey is the locally free OZy -module Ey = E ⊗ OZy .

Moreover, we say that E satisfies the even (odd resp.) WIT condition if Rjf∗Er = 0for all odd (resp. even) j, or that it satisfies the even (resp. odd) IT condition ifHj(Zy, Ey) = 0 for all y ∈ Y and all odd (resp. even) j. 4

We need the following technical result:

Lemma 5.30. Let Y be a complex manifold. Then C∞Y is a faithfully flat sheaf ofOY -modules.

Proof. The proof is easy but quite dull. Since a flat morphism of local rings isfaithfully flat, one has only to prove that the local ring C∞y is flat over (OY )yfor every point y ∈ Y . A result by Malgrange [207] asserts that C∞y is flat overthe subring of germs of (complex-valued) real analytic functions. That subringis isomorphic with the subring of convergent series Cz1, . . . , zn, z1, . . . , zn. The


problem then reduces to proving that Cz1, . . . , zn, z1, . . . , zn is flat over (OY )y =Cz1, . . . , zn. If Ct1, . . . , tn is the ring of convergent series in a neighborhood ofthe origin in Rn, we have to prove that Ct1, . . . , tn, tn+1 is flat over Ct1, . . . , tn.Now, there is a chain of ring morphisms

Ct1, . . . , tn → Ct1, . . . , tn[tn+1]

→ Ct1, . . . , tn[tn+1](tn+1) → Ct1, . . . , tn, tn+1 .

The first morphism is obviously flat, the second is the localization by theideal generated by tn+1, so it is flat as well. Let us notice that if A is a localNoetherian ring, and A is the completion of A in the topology of the maximalideals, then A is flat over A, hence is faithfully flat. Now, the third morphismis flat because it is an immersion of local Noetherian rings that induces an iso-morphism C[[t1, . . . , tn, tn+1]] ∼→ C[[t1, . . . , tn, tn+1]] between their completions inthe topology of maximal ideals.

Theorem 5.31. Let f : Z → Y be a proper morphism of complex manifolds. Aholomorphic bundle E → Z has a relative holomorphic structure given by Er =f∗hC∞Y ⊗OZ E, and one has:

1. The holomorphic higher direct images Rif∗E are coherent sheaves of OY -modules, and the natural map Rif∗E → Rif∗Er induces an isomorphism ofC∞Y -modules

C∞Y ⊗OY Rif∗E ∼→ Rif∗Er = Rif∗(f∗hC∞Y ⊗OZ E) .

2. E is WITi if and only if its holomorphic higher direct images vanish forj 6= i,

Rjf∗E = 0, for every j 6= i .

Let us assume that f : Z → Y is flat as a morphism of complex manifolds.

3. For any i, E is ITi if and only if it is both WITi and the only nonvanishingholomorphic higher direct image Rif∗E is a locally free OY -module.

4. In particular, E is IT0 if and only if it is WIT0.

Proof. The coherence of the holomorphic higher direct images for a proper mor-phism is Grauert’s semicontinuity theorem [130]. Since C∞Y is a flat OY -module(Lemma 5.30), the formula in (1) is a direct adaptation of the algebraic projectionformula [229]. Property (2) is a consequence of (1) and of the faithful flatness ofC∞Y as an OY -module (Lemma 5.30). Part (3) and (4) follow now straightforwardlyfrom Grauert’s cohomology base change theorem.

Remark 5.32. Points 1, 2, 3 in the previous theorem hold true if WITi is replacedby even or odd WIT, and ITi by even or odd IT. 4


5.3.3 Relative Dirac operators

Let us assume that f : Z → Y is a proper holomorphic submersive morphism,and that the bundle Ω1

Z/Y has a relative Kahler structure, i.e., it has a Hermitianmetric such that the corresponding 2-form in Ω2

Z/Y is closed under the relativeexterior differential.

We define locally free C∞Z -modules

Σ− =⊕k odd

Ω0,kZ/Y , Σ+ =

⊕k even

Ω0,kZ/Y , Σ = Σ+ ⊕ Σ− .

The Hermitian metric on Ω1Z/Y induces Hermitian metrics on the bundles Σ±.

Let now E → Z be a relatively holomorphic vector bundle with a Hermitianstructure. By considering the relative Dolbeault operator as a morphism

∂E/Y : f∗(E∞ ⊗C∞Z Σ)→ f∗(E∞ ⊗C∞Z Σ) ,

we define a relative Dirac operator

D = ∂E/Y + ∂∗E/Y : f∗(E∞ ⊗C∞Z Σ+)→ f∗(E∞ ⊗C∞Z Σ−) .

Theorem 5.33. Let E → Z be a relatively holomorphic vector bundle with a Hermi-tian structure, and let D be the corresponding relative Dirac operator. If E satisfiesthe odd IT condition, then kerD = 0. As a consequence, −ind(D) is a vector bun-dle and −ind(D) = cokerD = kerD∗. Moreover, there is a natural isomorphismof C∞Y -modules

−ind(D) = cokerD ∼→⊕odd i

Rif∗Er .

Proof. Since the higher direct images of Er are computed by the direct image ofthe relative Dolbeault complex (cf. Proposition 5.28), a direct calculation showsthat kerD =

⊕even iR

if∗Er = 0, thus proving the first claim. Analogously, onehas

kerD∗ = ker ∂E/Y ∩ ker ∂∗E/Y∼→⊕odd i

Rif∗Er .

Corollary 5.34. If, in addition to the hypotheses of Theorem 5.33, f : Z → Y isflat, and E is holomorphic, then the bundle −ind(D) admits a natural holomorphicstructure. Indeed,

⊕odd iR

if∗E is a locally free OY -module of finite rank, andthe natural map

⊕odd iR

if∗E →⊕

odd iRif∗Er induces an isomorphism of C∞Y -

modules ⊕odd i

Rif∗E ⊗OY C∞Y ∼→⊕odd i

Rif∗Er ∼→ − ind(D) . (5.10)


An analogous result is obtained by replacing “odd” with “even” and invertingthe roles of kerD and cokerD.

Proof. For every y ∈ Y , let ψiy : Rif∗E ⊗OY Oy → Hi(Zy, Ey) be the naturalmap. Since Hi(Zy, Ey) = 0 for even i, Grauert’s semicontinuity theorem impliesthat Rif∗E = 0 for even i, and then the morphisms ψiy are surjective for eveni. Cohomology base change implies that every ψiy is an isomorphism for odd i,and the for odd i, the sheaves Rif∗E are locally free. The isomorphism (5.10) nowfollows.

5.3.4 Kahler Nahm transforms

By means of the techniques developed in the previous section, one can define ahigh-dimensional version of the Nahm transform introduced in Section 5.2. As aconsequence, we also establish the compatibility between the Fourier-Mukai andNahm transforms for Kahler surfaces.

Let X, Y be compact Kahler manifolds. Fix on Z = X×Y a Hermitian vectorbundle Q equipped with a connection ∇Q whose curvature is of type (1, 1). Thus∇Q induces a holomorphic structure on Q, and let Q → Z denote its locally freesheaf of holomorphic sections. We shall denote as usual by πX , πY the projectionsof X × Y onto its factors.

Moreover, let (E,∇) → X be a Hermitian vector bundle with a connectionwhose curvature is of type (1, 1) on X. Again, ∇ induces a holomorphic structureon E. We shall again denote by E the corresponding sheaf of holomorphic sections.

Let us assume further that the Hermitian vector bundle π∗XE ⊗ Q satisfiesthe odd IT condition with respect to the projection πY . Then, as we saw in The-orem 5.33, minus the index of the relative Dirac operator D associated with thesedata is a holomorphic vector bundle on Y , which we denote E. Moreover, weknow that the sheaf E of holomorphic sections of E is isomorphic with the sheaf⊕

odd iRiπY ∗(π∗XE ⊗ Q). In this context, we have:

Proposition 5.35. Let X, Y be compact Kahler manifolds and let ΦQX→Y be theFourier-Mukai functor with kernel Q. Then the sheaf E of holomorphic sections ofE is isomorphic to ΦQX→Y (E).

The geometric data we have fixed at the outset also induce a Hermitianmetric and a compatible Chern connection ∇ on the holomorphic bundle E; clearlythe curvature of ∇ is of type (1,1). The pair (E, ∇) will be called the KahlerNahm transform of (E,∇). A notation for this transform which is in line withour notation for the integral functors of Chapter 1 is KNQ

X→Y (E,∇). It can bethought as a map from the space of gauge equivalence classes of connections onE → X whose curvature are of type (1,1) to the same space on E → Y .


We examine now this construction more closely. We claim that the inducedChern connection ∇ can also be obtained via a projection formula.

Indeed, for every y ∈ Y let Qy = Q|X×y and let ∇Qy be the connectioninduced on it. Moreover let us define bundles S±, S by

S− =⊕k odd

Ω0,kX , S+ =

⊕k even

Ω0,kX , S = S+ ⊕ S− .

We have of course S+ ' (Σ+)|X×y for all y ∈ Y , etc., where the Σ’s are the bun-dles defined in Section 5.3.3. Note that the Kahler metric of X induces Hermitianmetrics on the bundles S± and S, as well as compatible connections on them. Bycoupling the connections ∇, ∇Qy and the induced connections on S±, we obtainconnections on the bundles E⊗S±⊗Qy and a family of (twisted) Dirac operators

Dy : Γ(E ⊗Qy ⊗ S+)→ Γ(E ⊗Qy ⊗ S−) (5.11)

which are no more than the specializations to the fibers of πY of the relative Diracoperator introduced in Section 5.3.1, twisted by the coupled connections ∇± onthe bundle π∗XE ⊗Q⊗Σ±. Here Γ denotes the spaces of global C∞ sections. Thespaces Γ(E ⊗Qy ⊗ S±) have natural inner products given by the Kahler and thevarious Hermitian metrics, so that they may be completed to Hilbert spaces H±y .Since the holomophic bundle E satisfies the odd IT condition, we have for everyy ∈ Y an exact sequence (cf. Eq. (5.6))

0→ Ey → H−yD∗y−−→ H+

y → 0 . (5.12)

The spaces H±y may be regarded as the fibers of vector bundles H± on Y of infiniterank and the exact sequence (5.12) is then an exact sequence of vector bundles

0→ E → H−D∗−−→ H+ → 0 .

The inner products in the spaces H−y induce Hermitian inner products in thefibers Ey so that the bundle E has a Hermitian fiber metric h. This also defines aprojector Π: H− → E.

Let us now come to the connection. On the bundles H± one can defineconnections ∇± according to the following covariant derivative rule:

∇±α (s) = ∇±αX

(s)

where s is regarded as a section of H± in the left-hand side, and as a section in(the Hilbert space completion of) Γ(π∗XE ⊗ Q ⊗ Σ±) on the right. α is a vectorfield on Y , and αX is its natural lift to Z = X × Y [43]. The connection ∇ is nowdefined as

∇ = (Π× Id) ∇− .

5.4. Nahm transforms on hyperkahler manifolds 173

The operators D∗y vary holomorphically with y. Then standard arguments (cf. [102,Theorem 3.2.8]) show that this connection is compatible both with the Hermitianmetric and the holomorphic structure of E, and therefore coincides with the Chernconnection of the holomorphic Hermitian bundle (E, h).

Now assume that X and Y fit into the framework of Section 5.2, i.e., dimX =2, Y is a connected component of the moduli space of instantons onX (with respectto the Kahler metric on X) and ∇ is anti-self-dual. Let (Q,∇Q)→ X × Y be theuniversal bundle with connection, as described in Section 5.2.2. We have (see [162,Theorem 3]):

Proposition 5.36. If (Q,∇Q)→ X×Y is the universal bundle with connection, asdescribed in Section 5.2.2, then its curvature is of type (1, 1).

Proof. The proof follows from some easy computations, by taking into accountthe explicit form of the curvature of the universal connection (see Section 5.1.2)and the complex structure of the moduli space Y .

As discussed in Section 5.2, if X has nonnegative scalar curvature, thenπ∗XE ⊗ Q satisfies the odd IT condition with respect to the projection πY . It isthen easy to see that the Kahler Nahm transform KNQ

X→Y (E,∇) of (E,∇) in factcoincides with (E, ∇), the Nahm transform of (E,∇) as discussed in Section 5.2.

5.4 Nahm transforms on hyperkahler manifolds

When the base manifold X has a hyperkahler structure, we may address theproblem of whether the Nahm transform maps instantons to instantons. As amatter of fact the presence of the hyperkahler structure will allow us to considera generalization of the notion of instanton (that of quaternionic instanton) whichmakes sense on a hyperkahler manifold of any dimension. We shall be able to seethat the Nahm transform preserves the property of being a quaternionic instanton,thus generalizing the property of preservation of the instanton condition on 4-torias proved in [230, 57, 263, 102].

5.4.1 Hyperkahler manifolds

Let us start by recalling the basic notions concerning hyperkahler manifolds. Gen-eral references about hyperkahler manifolds are [262, 40, 147]

We recall that a 4k-dimensional Riemannian manifold (M, g) is said to bea quaternionic Kahler manifold (resp. a hyperkahler manifold) if its holonomygroup is contained in the group Sp(k) Sp(1) = (Sp(k)×Sp(1))/Z2 (resp. Sp(k)). A


hyperkahler manifold may be alternatively defined as Riemannian manifold (X, γ)equipped with three complex structures I, J , K that fulfil the following properties:

1. they satisfy the algebra of the quaternions, i.e.,

IJ = K, KI = J, JK = I, I2 = J2 = K2 = −Id ;

2. each of the complex structures I, J , K makes the Riemannian manifold(X, γ) into a Kahler manifold.

Let us denote by ωI , ωJ , ωK the three corresponding Kahler forms. One easilychecks that the 2-form ΩI = ωJ + iωK is holomorphic with respect to the complexstructure I, and is therefore a holomorphic symplectic form. As a consequence,the canonical bundle of X equipped with the holomorphic structure I is trivial.Analogous statements are obtained by cyclically permuting the complex structuresI, J , K. One should also notice that if a, b, c are real numbers such that a2 + b2 +c2 = 1, then aI + bJ + cK is a complex structure which again makes (X, γ) intoa Kahler manifold. Therefore any hyperkahler manifold comes with a family ofKahlerian structures parameterized by a 2-sphere S2. We call this the hyperkahlerfamily of complex structures of X.

5.4.2 A generalized Atiyah-Ward correspondence

In order to study the Nahm transform for hyperkahler manifolds, we shall need ageneralization of the classical Atiyah-Ward correspondence, which relates instan-tons on S4 to holomorphic vector bundles on the complex projective 3-space P3,satisfying suitable properties [20, 16]. This will rely on the treatment of [27], whichin turn mainly (but not only) reproduces results in [262, 40, 208, 236].

We shall start by introducing the twistor space Z of a (connected) quater-nionic Kahler manifold X, following [147]. Let P be holonomy bundle of X, i.e.,the principal bundle obtaining by reducing the structure group of the bundle oflinear frames of X to the holonomy group G [185, 40]. The space Z can be definedas the sphere bundle of the associated bundle W = P ×G sp(1), where G acts onthe Lie algebra sp(1) by its adjoint action. (To be more precise, note that G isa subgroup of Sp(k) Sp(1). The Lie algebra of the latter group is isomorphic tosp(k)⊕sp(1), hence one can restrict the adjoint action of G to sp(1); one can checkthat sp(1) is preserved by such action.) Any orthonormal basis of local sectionsof W pointwise defines a triple Ii of complex structures in the tangent spacesTxX, which satisfy the quaternionic algebra

Ii Ij = −δijId + εijkIk.


In terms of this trivialization, one makes the identifications u = (x, z) and TuZ =TxX ⊕ TzP1, and the tangent space TuZ has the complex structure(

1− zz1 + zz

I1 +z + z

1 + zzI2 + i

z − z1 + zz

I3, I0

), (5.13)

where I0 is the complex structure of P1. The resulting almost complex structureof Z is integrable (if k = 1, one needs the additional assumption that M is halfconformally flat, i.e., the self-dual part of the Weyl 2-form must vanish, cf., e.g.,[40]).

According to (5.13), the points u ∈ Zx = p−1(x) parameterize complexstructures in TxX: the complex structure labeled by u is the one induced bythe projection p∗ : TuZ → Tp(u)X. If k = 1 all complex structures of TxM arerecovered by varying u in Zx, while for k > 1 this is not anymore true.

On the twistor space there is a naturally defined antihomolorphic involutionτ : Z → Z which preserves the fibers of p. Its differential acts as

τ∗(α, β) = (α, ι∗(β)),

where ι is the antipodal map ι : P1 → P1. This implies the identities

τ ∂ = ∂ τ, τ ∂ = ∂ τ. (5.14)

The endomorphism∑3i=1 Ii ⊗ Ii of Λ2T ∗xX has real eigenvalues 3 and −1,

and correspondingly Λ2T ∗xX splits into the eigenspaces [208]

Λ2T ∗xX = (e1)x ⊕ (e2)x .

The space (e1)x ' sp(n) admits the further identification

(e1)x '⋂u∈Zx

Λ1,1u T ∗xX (5.15)

where Λ1,1u T ∗xX is the space of 2-forms of type (1,1) with respect to the complex

structure in TxX parameterized by the point u ∈ Zx.

Definition 5.37. A rank n quaternionic instanton on quaternionic Kahler manifoldsX is a pair (E,∇), with E a rank n complex smooth vector bundle on X, and ∇ aconnection on E such that its curvature at x ∈ X takes values in (e1)x⊗End(Ex).

4

By using the Hitchin-Kobayashi correspondence (Section 5.1.3) and a resultby Verbitsky [290], one can easily prove the following characterization of quater-nionic instantons on hyperkahler manifolds as “hyperstable bundles.”


Proposition 5.38. A complex vector bundle on a hyperkahler manifold X is anirreducible quaternionic instanton if and only it is a holomorphic stable bundle ofdegree zero with respect to any Kahler structure in the hyperkahler family of X.

Remark 5.39. Contrary to what happens in the case of complex dimension two,in the higher dimensional case a line bundle on a hyperkahler manifold is notnecessarily a quaternionic instanton. Indeed, its first Chern class may fail to beorthogonal to all the three basic Kahler forms of X. 4

Let us now study the generalized Atiyah-Ward correspondence for quater-nionic instantons. We start by considering the Hermitian case, namely, we assumethat E is a complex vector bundle on a quaternionic Kahler manifold, and that Eis equipped with a Hermitian metric h and a quaternionic instanton ∇ which iscompatible with the Hermitian metric h. Because of (5.15), the pullback connec-tion p∗∇ induces a holomorphic structure on the pullback bundle W = p∗E (sothat (p∗∇)0,1 = ∂W ). Then W is holomorphically trivial along the fibers of p. Wedefine a bundle morphism σ : W →W ∗ by letting

σ(s1)(s2) = (p∗h)(s1, τ′(s2)) ,

where the antiholomorphic bundle automorphism τ ′ : W → W is the lift of theinvolution τ . Then σ is an antilinear antiholomorphic bundle isomorphism coveringτ , and induces a positive definite Hermitian form on the spaces H0(Zx,Wx).

The map σ is a real positive form according to Atiyah’s terminology [16].Thus, with any quaternionic instanton on X we may associate a holomorphicvector bundle on Z, holomorphically trivial along the fibers of p, carrying a positivereal form.

Let us now describe the inverse correspondence. Let W be such a bundle onZ. By the triviality requirement, there is a bundle E on X such that W = p∗E.Since two points e1, e2 ∈ Ex may be regarded as elements in H0(Zx,Wx), the realpositive form σ induces a Hermitian structure h on E.

Let ∇ be the connection on W uniquely determined by the compatibilitywith the holomorphic structure of W and the Hermitian structure induced on W

by σ (which obviously coincides with p∗h); i.e., ∇ is the Chern connection of theHermitian bundle (W,p∗h). The connection ∇ descends to a connection on E ifand only if the curvature F∇ is horizontal with respect to p. Let us briefly showthat this is indeed the case. Let Ix be the ideal of the fiber Zx. As in [16] one canproduce an isomorphism

H0(Zx, F ⊗OZ/I2x) ∼→ H0(Zx, F ⊗OZ/Ix) , (5.16)

where F ⊗OZ/Ix may be identified with the sheaf of holomorphic sections of Fx.

Due to the isomorphism (5.16), we may choose in a neighborhood U of Zxa basis si of holomorphic sections of F which restricted to Zx yields a unitary


basis of holomorphic sections of Fx. If ωij is the matrix-valued connection 1-formof ∇, we have

ω0,1ij ∈ I

2xΩ0,1(U), ωij + ωji ∈ I2

xΩ1(U), (si, sj)− δij ∈ I2x .

By computing the local curvature forms we see that the curvature is horizontal.

The induced connection ∇ on E is compatible with the Hermitian metrich, and we need only to show that at each point x ∈ M the curvature F∇ of ∇takes values in (e1)x ⊗ EndEx. Now, the two-form (F∇)u is of type (1,1) at anypoint u ∈ Zx, so that the two-form (F∇)x at x is of type (1,1) with respect to thecomplex structure on TxM parameterized by u. Since this happens for all u ∈ Zx,Equation (5.15) implies that (R∇)x lies in (e1)x.

We have therefore proved the following result.

Theorem 5.40. There is a one-to-one correspondence between the following objects:

1. gauge equivalence classes of rank r Hermitian quaternionic instantons on aquaternionic Kahler manifold X;

2. isomorphism classes of rank r holomorphic vector bundles on the twistorspace Z of X, holomorphically trivial along the fibers of Z, carrying a positivereal form.

Remark 5.41. In [236] a similar result was proved, but instead of the condition ofholomorphic triviality along the fibers of p, it is assumed there that the holomor-phic Hermitian bundle (F, k) restricted to the fibers is flat. The equivalence of thetwo constructions is easily established. 4

For the sake of completeness we also treat the non-Hermitian case. Clearly,if (E,∇) is a quaternionic instanton on X, we may construct on the twistor spaceZ a holomorphic bundle W , which is holomorphically trivial along the fibers ofthe projection p. We show how to recover the quaternionic instanton (E,∇) fromsuch data on Z (in particular now we have no real form at our disposal).

We define a differential operator D : Γ(W )→ Γ(W ⊗ Ω1,0Z ) by letting

D(s) = τ ′ ∂W τ ′(s).

Due to the identities (5.14), this operator satifies the Leibniz rule

D(fs) = ∂f ⊗ s+ fD(s),

so that ∇ = D+ ∂W is a connection on W , compatible with its complex structure(in particular, since D2 = ∂2

W = 0, the curvature F∇ of ∇ is of type (1,1)). By thesame argument as in the previous case, ∇ descends to a connection ∇ on E, suchthat (E,∇) is a quaternionic instanton.

One has therefore:


Theorem 5.42. There is one-to-one correspondence between the following objects:

1. gauge equivalence classes of quaternionic instantons of rank r on a quater-nionic Kahler manifold X;

2. isomorphism classes of holomorphic vector bundles of rank r on the twistorspace Z of X which are holomorphically trivial along the fibers of Z.

5.4.3 Fourier-Mukai transform of quaternionic instantons

We introduce now a version of the Fourier-Mukai transform which is apt to studyquaternionic instantons in that it uses in an essential way the twistor space. Werestrict here to hyperkahler manifolds. There are several reasons for doing so: onhyperkahler manifolds we can make full use of techniques from complex geometry;the product of two hyperkahler manifolds has a natural hyperkahler structure;and the twistor space ZX of a hyperkahler manifold X is isomorphic to X ×P1 asa smooth manifold, a condition which we shall need in the sequel.

This Fourier-Mukai transform will map quaternionic instantons on a com-pact hyperkahler manifold X to quaternionic instantons on a second hyperkahlermanifold Y . The product X × Y carries a natural hyperkahler structure, namely,the only one compatible with the isomorphism of complex manifolds

ZX×Y ' ZX ×P1 ZY .

One has a commutative diagram

ZX

p1

ZX×Yt1oo t2 //

q

ZY

p2

X X × Y

π1oo π2 // Y

where the horizontal arrows are holomorphic morphisms while the vertical onesare just smooth. Moreover we denote by

ρ1 : ZX → P1, ρ2 : ZY → P1, $ : ZX×Y → P1

the holomorphic projections of the twistor spaces onto the projective line.

Let now E be the sheaf of holomorphic sections of a Hermitian quaternionicinstanton (E,∇) on X, and let (P,∇∇∇) be a Hermitian quaternionic instanton onX×Y . Let E and P denote the sheaves of holomorphic sections of the bundles p∗1Eand q∗P , respectively. After setting W = π∗E ⊗ P , and endowing the bundle Wwith the product connection, we denote by Wz the sheaf of holomorphic sections


of W determined by the the complex structure Iz on X, where z ∈ P1. If one setsW = t∗1E ⊗ P, there is an identification of holomorphic vector bundles W|$−1(z) =Wz.

We assume now that (E,∇) satisfies one of the even or odd IT conditions(cf. Definition 5.29; when this happens, we say that (E,∇) “is IT”). One shouldnotice here that this does not depend on the choice of a complex structure in thehyperkahler family of X; in fact, the cohomology groups Hi(Xz×y,Wz |Xz×y)do not depend on the choice of z (here Xz is X with the complex structure z, i.e.,Xz = ρ−1(z)). This is proved in [290].

According to the results of the previous sections, the smooth vector bun-dle underlying the holomorphic bundle ⊕iRiπ2∗(Wz) may be identified with thebundle ±ind(Dz), where Dzz∈P1 is a family of Dirac operators in the senseof Atiyah-Singer’s index theorem; the plus (minus) sign holding when the even(odd) IT condition is satisfied. All the smooth Hermitian bundles ±ind(Dz) areisomorphic, since all the operators D∗z Dz have isomorphic (co)kernels, and willbe denoted by ±ind(D). Furthermore, this bundle is endowed with a Hermitianconnection, which is compatible with all complex structures on X.

We notice that, if u ∈ ZY , then t−12 (u) = Xz, with z = ρ2(u); therefore, we

make the identifications W|t−12 (u) =W|Xz×p2(u). We then have:

Lemma 5.43. If (E,∇) is IT, then W is IT with respect to t2 : ZX×Y → ZY .

Proof. It suffices to observe that Hi(t2−1(u), Wt2−1(u)) = Hi(Xz,W|Xz×u).

We consider the following commutative diagram:

Xz × Yzjz //

π2

ZX×Y

t2

$ // P1

Yziz // ZY

ρ2

<<xxxxxxxx

where iz and jz are the natural holomorphic embeddings.

The square on the left is Cartesian; indeed, Xz × Yz = $−1(z) and Yz =ρ−1

2 (z), so that ZX×Y ×ZY Yz = t2−1(Yz) = Xz×Yz. Therefore, although iz is not

flat, we have natural maps µi : i∗zRit2∗W → Riπ2∗Wz (notice that j∗zW ' Wz).

We remark that the smooth bundle underlying ⊕iRit2∗(W) is ±ind(D),where D is a family of Dirac operators that, when restricted to Xz, coincideswith Dz. Thus we have proved:

Lemma 5.44. The natural morphism ⊕iµi is an isomorphism of holomorphic Her-mitian vector bundles i∗z ⊕i Rit2∗W ' ⊕iRiπ2∗Wz.


Let us denote G = ⊕iRit2∗W. For any z ∈ P1, one has an isomorphism ofsmooth bundles (iz p2)∗G∞ ' p2

∗(±ind(D)). By applying the Riemann-Rochformula, it is easy to check that the restriction of G∞ to each P1-fiber is trivial(indeed, c1(G∞ |y×P1) = 0 for any y ∈ Y ). In this way, we see that G∞ isthe pullback of a smooth bundle on Y , which can be identified with ±ind(D).The pullback of the Hermitian connection on ±ind(D) is a Hermitian connectionon G∞, which is compatible with the holomorphic structure of G by virtue ofLemma 5.44. This implies that G is holomorphically trivial along the fibers of thetwistor projection p2. In particular, the smooth vector bundle ±ind(D) carries thestructure of a quaternionic instanton on Y .

Summing up, we have proved:

Theorem 5.45. Let (E,∇) be a quaternionic instanton on X which satisfies oneof the IT conditions. Then ⊕iRiπ2∗(π∗1E ⊗P) is the sheaf of holomorphic sectionsof a quaternionic instanton on Y .

5.4.4 Examples

The only compact hyperkahler surfaces are the complex 2-tori and the K3 surfaces(see, e.g., [40]). The case of complex 2-tori was considered in Section 5.2.4. Anotherexample is provided by the strongly reflexive K3 surfaces of Chapter 4. Usingthe Hitchin-Kobayashi correspondence, the “dual” reflexive surface X, which isa moduli space of µ-stable locally free sheaves, may be identified with a modulispace of instantons (note indeed that the locally free sheaves parameterized by Xhave zero degree). According to results given in [200], the two moduli spaces canbe identified as complex manifolds. Moreover, the universal instanton bundle Qon the product X × X admits a holomorphic structure, since the curvature of theuniversal connection∇∇∇ is of Hodge type (1,1), and its sheaf of holomorphic sectionmay be identified with the universal sheaf that we have constructed in Chapter 4.Now, every choice of complex structure in the hyperkahler family of X induces acomplex structure in X, and in this way we obtain a hyperkahler structure on theproduct X×X. Arguments given in [162] imply that the pair (Q,∇∇∇) is a Hermitianquaternionic instanton. Therefore we have all the ingredients for building up ahyperkahler Fourier-Mukai transform on X. Our construction however shows thatthis coincides with the Fourier-Mukai transform built in Chapter 4, and under thiscorrespondence, Theorem 5.45 coincides with Proposition 4.66. (Note that in thiscase, in the statement of Theorem 5.45 we need only to say “instanton” insteadof “quaternionic instanton” since dimX = 2).

We may consider the hyperkahler Fourier-Mukai transform for hyperkahlertori of higher dimension (i.e., complex tori of even dimension). However we canremark here that, contrary to what happens in complex dimension 2, in higher


dimensions the Fourier-Mukai transform is not well behaved on stable bundles.Let T1, T2 be hyperkahler tori. Consider for instance a stable rank-two vectorbundle E on T1 with vanishing first Chern class, and a flat line bundle L on T2.Since E is IT1 and L is WIT2 by a Kunneth formula [134, 6.7.8.1], one shows thatthe bundle F = E L on T1 × T2 is WIT3, and by considering its restrictions tosubsets of the type T1×x2 and x1×T2, one proves that it is stable. However,the Fourier-Mukai transform W has rank 1

2c2(E)c1(L)2 = 0, i.e., it is a torsionsheaf.


Physics literature. There is an extensive physical literature relating Nahm trans-form and fundamental problems in physics, like quark confinement in QCD andstring dualities. For the reader interested in these issues, we recommend for in-stance [121] (among other papers by Pierre van Baal) for the relevance of Nahmtransform in QCD on the lattice and [88, 92, 172, 292] for the relations betweenNahm transform and string theory. In [13] a version of Nahm transform for in-stantons over noncommutative 4-tori was introduced.

Nahm transform in K-Theory. We would like to notice that the construction pre-sented above is essentially topological, in the sense that its main ingredient issimply index theory. All the geometric structures used in Section 5.2 (spin struc-ture, positivity of scalar curvature, hyperkahler metric, etc.) were needed eitherbecause a particular differential operator was used (i.e., the Dirac operator), orbecause we selected those objects (i.e., anti-self-dual connection over hyperkahlermanifolds) that yielded very particular transforms (anti-self-dual connections).One can conceive, for instance, a similar construction either based on a differentpseudodifferential elliptic operator, other than the Dirac operator, or allowing forclasses in K(T ), rather than actual vector bundles over the parameter space.

Translation invariant instantons. There exists an ample literature on Nahm trans-forms of translation invariant instantons. Roughly speaking, these Nahm trans-forms yield a one-to-one correspondence between instantons on R4 invariant underthe action of a lattice Λ and Λ∗-invariant instantons on the dual R4. Let us citesome cases:

• The “trivial” case Λ = 0 is closely related to the celebrated ADHM con-struction of instantons, as described by Donaldson and Kronheimer [102]; inthis case, Λ∗ = R4, and an instanton on R4 corresponds to some algebraicdata (ADHM data). This has been worked out by Corrigan and Goddard[90].

• Λ = R corresponds to monopoles; here, Λ∗ = R3, and the transformed


object is, for SU(2) monopoles, an analytic solution of Nahm’s equationsdefined over the open interval (−1, 1) and with simple poles at the endpoints.This case was extensively studied by several authors, including Hitchin [146],Donaldson [98], Hurtubize and Murray [152], and Nakajima [231].

• The case Λ = R3 was treated by Szabo in his thesis [272]

• For Λ = Z4 we have the Nahm transform of Schenk [263], Braam and vanBaal [57], and Donaldson and Kronheimer [102], defining a correspondencebetween instantons over two dual 4-dimensional tori, as discussed in Section5.2.4.

• Λ = Z corresponds to the so-called calorons, studied by Nahm [230], vanBaal [287] and others (see [237] and the references therein and [86]); thetransformed object is the solution of Nahm-type equations on a circle.

• The case Λ = Z2 (doubly periodic instantons) has been analyzed in greatdetail in [41, 163, 165, 164]; here, Λ∗ = Z2 × R2, and the Nahm transformgives a correspondence between doubly-periodic instantons and certain tamesolutions of Hitchin’s equations on a 2-torus.

• Λ = R × Z gives rise to the periodic monopoles considered by Cherkis andKapustin [88]; in this case, Λ∗ = Z× R, and the Nahm dual data are givenby certain solutions to Hitchin’s equations on a cylinder.

• More recently, the case Λ = Z3 (spatially periodic instantons) has beenstudied by Charbonneau [85]; the transformed object is a singular monopoleon a 3-torus. Previous work in that case had been done by van Baal [287, 288].

Instantons on ALE spaces. Asymptotically locally Euclidean spaces (ALE spaces)are hyperkahler manifolds that are obtained as resolutions of singularities of quo-tients C2/Γ, where Γ is a discrete subgroup of SU(2), acting on C2 in the standardway [189]. Instantons on ALE spaces have been first studied by Kronheimer andNakajima [190] and have since then received a lot of attention. A Nahm transformfor instantons on ALE spaces has been constructed by Bartocci and Jardim [30].

Other examples. The Nahm transform has been applied to study vortexes (in par-ticular, some holomorphic triples over elliptic curves) by Garcıa-Prada, HernandezRuiperez, Pioli and Tejero [122]. A related construction is the Nahm transform forHiggs bundles as defined in [165] (Bonsdorff and Bartocci-Biswas studied Fourier-Mukai tranforms for Higgs bundles in [23] [53]). Another significant application tothe integer quantum Hall effect is due to Tejero [276].

Chapter 6

Relative Fourier-Mukai functors

Introduction

In this chapter we offer a quite comprehensive study of the relative Fourier-Mukaifunctors. We consider (proper) morphisms of algebraic schemes X → B, Y → B,and use an element in the derived category of the fibered product X ×B Y as akernel to define an integral functor from the derived category of X to the derivedcategory of Y . This generalizes what we have already seen in Chapter 1 when themorphisms X → B, Y → B are projections onto a factor of a product.

We start by giving some general properties, in particular base change for-mulas, and a first example: Mukai’s relative transform for Abelian schemes [226].We then move over to the case of elliptic fibrations (where by “elliptic fibration”we mean a proper flat morphism whose fibers are Gorenstein curves of arith-metic genus 1). Our treatment here may be divided in two parts, where we areable to achieve different degrees of generality in different directions. Indeed, weconsider at first Weierstraß fibrations, leaving the dimension of the base schemearbitrary. Given a Weierstraß fibration X → B, we construct a Poincare sheaf onthe fibered product X ×B X and use this as a kernel to define a relative Fourier-Mukai transform Db(X) → Db(X). We use this to provide a direct constructionof the Altman-Kleiman compactified relative Jacobian of a Weierstraß fibration(proving that it is actually isomorphic to the original fibration). We also study thisFourier-Mukai transform in some detail, in particular computing the topologicalinvariants of the transforms for elliptic surfaces and elliptic Calabi-Yau threefolds.

The second approach we study in connection with elliptic fibrations is due

While this works only in the case of elliptic surfaces, it has some advantages; for

183Progress in Mathematics 276, DOI: 10.1007/b11801_6,C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics,

to Bridgeland, and does not require the fibration to be of the Weierstraß type.


184 Chapter 6. Relative Fourier-Mukai functors

instance, it may be used to study Fourier-Mukai partners of elliptic surfaces, aswe shall do in Chapter 7.

Then, we prove a generalization of Atiyah’s characterization of semistablesheaves on elliptic curves to the case of Weierstraß (possibly singular) ellipticcurves. Moreover we prove the preservation of (semi)stability under the Fourier-Mukai transform for such curves. This partly relies on our paper [29], where a resultof this type was proved. This allows us to characterize the category of semistablesheaves on Weierstraß elliptic curves.

These results are then generalized to Weierstraß fibrations, in particular de-scribing all moduli spaces of relatively semistable sheaves, assuming that the basescheme B is normal. Some of the results given along this line in Section 6.4 are new.Partial results were contained in [29, 145], and in papers by Friedman, Morganand Witten, by Donagi, by Bridgeland and by Yoshioka [114, 94, 60, 293].

The next topic we consider is the spectral cover construction. In particularwe study how stable bundles on an elliptic fibration (in dimension 2 and 3) may bebuilt out of spectral data. Spectral covers are particularly interesting for relativelysemistable torsion-free sheaves of degree 0. In this case the spectral cover is afinite cover of the base, of degree equal to the rank of the sheaf. The Fourier-Mukai transform of such relatively semistable sheaf turns out to be a rank onetorsion-free sheaf on the spectral cover. Due to the invertibility of the transform,the sheaf may be recovered from its spectral data (the pair formed by the spectralcover together with a rank one torsion-free sheaf on it).

These results may be used to study sheaves on the total space of a Weierstraßfibration which are (semi)stable in an “absolute” sense. The analysis is limited hereto elliptic surfaces and elliptic Calabi-Yau threefolds since we need to know howthe Fourier-Mukai transform acts on the topological invariants of the sheaves.We prove some instances of the preservation of this kind of (semi)stability underthe Fourier-Mukai transform. This analysis is also useful to provide examples ofabsolutely µ-stable sheaves on elliptic Calabi-Yau threefolds, which are obtainedout of spectral data. This is relevant to the construction of compatifications of theheterotic string.

In this chapter we assume that the ground field k has characteristic zero; thisallows us to apply Proposition 1.27 about fully faithful integral functors. However,most of Section 6.1 is true in arbitrary characteristic.

6.1 Relative integral functors

Let p : X → B, q : Y → B be proper morphisms of algebraic varieties. We denoteby πX , πY the projections of the fiber product X ×B Y onto its factors and by

6.1. Relative integral functors 185

ρ = p πX = q πY the projection of X ×B Y onto the base scheme B. We havea Cartesian diagram

X ×B YπX

zzvvvvvvvvvπY

##HHHHHHHHH

ρ

Xp

$$IIIIIIIIII Yq

zzvvvvvvvvvv

B

. (6.1)

Given a “relative kernel” K• in the derived category D−(X ×B Y ), the relativeintegral functor with kernel K• is the functor Φ: D−(X)→ D−(Y ) given by

Φ(E•) = RπY ∗(Lπ∗XE•L⊗K•) .

This can be regarded as an integral functor with kernel j∗K• in the derived categoryD−(X × Y ), where j : X ×B Y → X × Y is the closed immersion of the fiberproduct. We can then apply all results about integral functors described in Chapter1 to relative integral functors. In particular, WITi and ITi notions introduced inDefinition 1.6 apply to this new situation.

Assume now that K• is of finite Tor-dimension as a complex of OX -modules.As j∗K• may fail to have this property, we cannot apply Proposition 1.4. Never-theless, we can modify the proof of that proposition to show that Φ is boundedand can be extended to a functor Φ: D(X)→ D(Y ) which maps Db(X) to Db(Y ).

As in the absolute case, the composition of two relative integral functors isobtained by convoluting the corresponding kernels. So, given two kernels K• inD−(X ×B Y ) and L• in D−(Y ×B Z) corresponding to relative integral functorsΦ and Ψ, the composition Ψ Φ has kernel in D−(X ×B Z)

L• ∗B K• = RπXZ∗(Lπ∗XYK•L⊗Lπ∗Y ZL•)

where the morphisms πXY , πXZ and πY Z are the projections of the fiber productX ×B Y ×B Z onto the fiber products X ×B Y , X ×B Z and Y ×B Z.

6.1.1 Base change formulas

As we saw in Section 1.2.1, what makes relative integral functors interesting istheir compatibility with base change. Let f : S → B be a morphism. For anymorphism g : Z → B (a scheme over B), we denote by gS : ZS = Z ×B S → S andfZ : ZS → Z the induced morphisms. The kernel K•S = Lf∗X×BYK

• gives rise to a


relative integral functor

ΦS : D−(XS)→ D−(YS)

ΦS(E•) = RπYS∗(Lπ∗XSE

•L⊗K•S) .

If the original kernel K• is of finite Tor-dimension as a complex of OX -modules,then K•S is of finite Tor-dimension as a complex of OXS -modules, so that ΦS isbounded, and for every f : S → B it can be extended to a functor ΦS : D(XS)→D(YS), mapping Db(XS) to Db(YS). In the rest of this section we assume indeedthat K• is of finite Tor-dimension as a complex of OX -modules.

The proof of the following base change compatibility result is analogous tothat of Proposition 1.8. Note that because of the flatness condition, base changein the derived category (Proposition A.85) can be applied. It is worth observingthat, if the morphism p : X → B is flat, there is no need to assume that the basechange morphism is flat (this fact is often neglected).

Proposition 6.1. Assume either that f : S → B or p : X → B is flat. For everyobject E• in Db(X) there is a functorial isomorphism

Lf∗Y Φ(E•) ' ΦS(Lf∗XE•)

in the derived category of YS.

Let us assume that p : X → B is flat. If E• ∈ Db(X), by denoting by jt theimmersions of both fibers Xt = p−1(t) and Yt = q−1(t) over a closed point t ∈ Binto X ×B Y , one has

Lj∗t Φ(E•) ' Φt(Lj∗t E•) . (6.2)

Whenever the morphism q : Y → B is flat, from the base change formula (Propo-sition A.85) we also have

jt∗Φt(G•) ' Φ(jt∗G•) (6.3)

for every G• ∈ D(Xt).

A straightforward consequence of Proposition 1.11 and Equation (6.2) is thefollowing result.

Corollary 6.2. Assume that p : X → B is flat, and let E• be an object in Db(X).Then the derived restriction Lj∗t E• to the fiber Xt is WITi for every t if and onlyif E• is WITi and Φi(E) is flat over B.

When we transform a complex that reduces to a single sheaf E on X, we haveΦi(E) = 0 for i > n = dim p+m0, where dim p is the maximum of the dimensionsof the fibers of p and m0 is the biggest index m such that Hm(K•) 6= 0. We havea result analogous to Corollary 1.9.

6.1. Relative integral functors 187

Corollary 6.3. Let p : X → B be a flat morphism and E be a sheaf on X. Thefunctor Φn is compatible with base change for sheaves, that is, one has

Φn(E)t ' Φnt (Et) ,

for every (closed) point t ∈ B, where Et = j∗t E. Moreover, if E is flat over B onehas:

1. for every (closed) point t in B there is a convergent spectral sequence

E−j,i2 (t) = TorOBj (Φi(E),Ot) =⇒ Ei−j∞ (t) = Φi−jt (Et) .

2. Assume that E is WITi and write E = Φi(E). Then for every t ∈ B thereare isomorphisms of sheaves over Xt

TorOBj (E ,Ot) ' Φi−jt (Et) , j ≤ i .

3. The restriction Et to the fiber Xt is WITi for every (closed) point t ∈ B ifand only if E is WITi and E = Φi(E) is flat over B. In that case the functorΦi is compatible with base change for sheaves, that is, (E)t ' Et for everypoint t ∈ B.

Proposition 6.4. Let p : X → B be a flat morphism and E be a sheaf on X flatover B. The set U of points in B such that the restriction Et of E to the fiber Xt

is WITi has a natural structure of open subscheme of B.

Proof. Given a point t ∈ B, we consider the flat base change Bt = SpecOB,t →B where OB,t is the local ring of B at the point t. By Proposition 6.1, the restric-tion Φj(E)Bt of Φj(E) to the fiber product Yt = Bt ×B Y is isomorphic toΦjBt(EBt). Then Corollary 6.3 applied to p : Xt = Bt ×B X → Bt impliesthat Et is WITi for a closed point t ∈ B, if and only if Φj(E)Bt = 0 for j 6= i andΦi(E)Bt is flat over Bt. By the generic flatness criterion [214, 22.B], the set ofthe points t ∈ B such that the two last conditions are fulfilled is open.

We are now going to apply Proposition 6.4 to the particular situation of a rela-tive integral functor induced by an ordinary integral functor Φ = ΦK

•

X→Y : Db(X)→Db(Y ), where X and Y are smooth connected proper varieties and K• is an ob-ject of Db(X ×Y ) of finite Tor-dimension as a complex of OX -modules. Following


Section 1.2.1, we consider the base change diagram

X ×X × Yπ12

xxpppppppppppπ13

&&NNNNNNNNNNN

ρ

X ×Xπ1

''NNNNNNNNNNNN X × Yπ1

xxpppppppppppp

X

,

where πij denote the projection onto the (i, j)-factor, and the relative integralfunctor ΦX from D−(X ×X) to D−(X × Y ) with kernel K•X = π∗23K•.

Proposition 6.5. The set U of points x in X such that the skyscraper sheaf Ox isWITi with respect to Φ has a natural structure of open subscheme of X.

Proof. We apply Proposition 6.4 to the integral functor ΦX : Db(X×X)→ Db(X×Y ) taking as E the structure sheaf O∆ of the diagonal in X ×X.

By Lemma 2.46, there is an irreducible component Z of the support of K•such that pX = πY |Z : Z → X is dominant.

Proposition 6.6. Assume that there is a (closed) point x ∈ X such that Φ(Ox) 'Oy[i] for some (closed) point y ∈ Y and some integer i. If Z is the normalizationof Z, the induced morphisms pX : Z → X and pY : Z → Y are birational. Thus,X and Y are K-equivalent (cf. Definition 2.47).

Proof. Since pX is dominant, dim p−1X (x) ≥ dimZ − dimX. Since p−1

X (x) = y,one has dimZ = dimX. Since X and Y have the same dimension (cf. Theorem2.38), we can apply Proposition 2.48 to conclude.

6.1.2 Fourier-Mukai transforms on Abelian schemes

An instance of a relative integral functor is provided by the Fourier-Mukai trans-form on Abelian schemes [226]; this has indeed been the first example of a sucha transform. An Abelian scheme p : X → B over a scheme B is a proper flatmorphism such that there exist morphisms of B-schemes

mX : X ×X → X, ιX : X → X, e : B → X

so that the relations described at the beginning of Section 3.1 are satisfied. Inanalogy with the absolute case, one proves the existence of an Abelian schemep : X = Pic0(X/B) → B which is a fine moduli space for line bundles whoserestrictions to the fibers of p have vanishing first Chern class. Universality implies

6.2. Weierstraß fibrations 189

the existence of a Poincare line bundle P on X ×B X, which we normalize byimposing that its restriction to the section e(B)×B X of the projection X×B X →X is trivial. For every closed point t ∈ B the restriction of P to the fiber Xt × Xt

of X ×B X coincides with the normalized Poincare bundle on Xt × Xt.

The line bundle P defines a relative integral functor Φ: D−(X) → D−(X)by letting

Φ(E•) = RπX∗(π∗XE• ⊗ P) .

Since P is a line bundle, the functor Φ maps Db(X) to Db(X). For the same reason,and using formula (C.12), one proves as in Proposition 1.13 that the relativeintegral functor from Ψ: Db(X)→ Db(X) defined by the kernel P∗ ⊗ π∗XωX/B [g]is a right adjoint to the functor Φ (here g is the relative dimension of the Abelianscheme p : X → B).

Proposition 6.7. The relative integral functor Φ is a Fourier-Mukai transform.

Proof. We know that the composition Ψ Φ: Db(X) → Db(X) is the relativeintegral functor with kernel given by the convolutionM• = (P∗⊗π∗XωX/B [g])∗BP,and that the composition Φ Ψ: Db(X)→ Db(X) is the relative integral functorwith kernel N • = P∗B (P∗⊗π∗XωX/B [g]). We first prove that Φ is fully faithful. ByRemark 1.21 it suffices to show that ΨΦ is fully faithful. Actually, we shall provethat it is an equivalence of categories. In view of the base change property given inProposition 6.1, for any point t ∈ B the composition Ψt Φt : Db(Xt) → Db(Xt)is the integral functor with kernel Lj∗tM•. By Theorem 3.2 the functor Φt is aFourier-Mukai transform, the composition Ψt Φt is isomorphic to the identityfunctor, and its kernel is isomorphic to O∆t

, where ∆t is the diagonal in Xt×Xt.Proposition 1.11 then implies that M• is isomorphic in the derived category to asheafM, flat over B, such that j∗tM' O∆t

for every point t ∈ B. Moreover,M istopologically supported on the image of the diagonal immersion δ : X → X×BX.Let us denote by L = δ∗M the restriction of M to the diagonal; we have anepimorphism M→ δ∗L → 0. The condition j∗tM' O∆t

implies that j∗t L ' OXtfor every t ∈ B and that M → δ∗L induces an isomorphism j∗tM ' j∗t δ∗L forevery point t ∈ B. Since M and δ∗L are flat over B, the morphism M→ δ∗L isan isomorphism as well. As a consequence, the composed functor Ψ Φ coincideswith the operation of tensoring by L, so it is an equivalence of categories.

A similar argument proves that ΦΨ is an equivalence of categories, and thatΨ is fully faithful as well. This implies that Φ is an equivalence of categories.

6.2 Weierstraß fibrations

Elliptic fibrations yield examples of relative Fourier-Mukai transforms that are ofgreat interest in view of their geometric and physical applications [28, 60, 29, 71,


82, 145, 6]. We shall adopt the following definition of elliptic fibration (not themost general).

Definition 6.8. Let B be an integral and projective scheme. An elliptic fibrationover B is a proper flat morphism of schemes p : X → B whose fibers are Gorensteincurves of arithmetic genus 1. 4

If X is smooth, the generic fiber of p is a smooth elliptic curve, but singu-lar fibers are allowed. The simplest nontrivial examples are provided by ellipticsurfaces.

Definition 6.9. A relatively minimal elliptic surface is an elliptic fibration p : X →B such that B is a smooth projective curve, X is smooth, and there are no (−1)-curves (i.e., rational curves C with C2 = −1) contained in the fibers. 4

Relatively minimal elliptic surfaces were classified by Kodaira [186], who de-scribed all types of singular fibers which may occur (the so-called Kodaira curves).Elliptic fibrations whose base is a smooth surface have been studied by Miranda[219], who showed that the configuration of singular fibers can be more complicatedthan in the case of elliptic surfaces.

We say that a sheaf over an elliptic fibration p : X → B is relatively torsion-free if it is flat over B and its restriction to every fiber is torsion-free. In ananalogous way one defines the notion of relative µS-(semi)stability (cf. DefinitionsC.3 and C.4).

If E• in Db(X) is a complex of finite Tor-dimension its relative degree is theintersection number

d(E•) = c1(E•) · f , (6.4)

where f ∈ Am(X) is the class of the generic fiber of p (here m = dimB). If F is asheaf on X flat over B, its relative degree is the degree of the restriction Ft to anyfiber Xt of p. The pair (rk(E•), d(E•)) (cf. Section 1.1 for the definition of rank inthe derived category) is called the relative Chern character of E•. If rk(E•) 6= 0,the rational number µ(E•) = d(E•)/ rk(E•) is the relative slope.

6.2.1 Todd classes

We now focus on a particular, though very important, kind of elliptic fibration.

Definition 6.10. A Weierstraß fibration is an elliptic fibration p : X → B such thatthe fibers of p are geometrically integral and there exists a section σ : B → X ofp whose image Θ = σ(B) does not contain any singular point of the fibers. 4

We notice that the singular fibers can have at most one singular point, eithera cusp or a simple node.


By cohomology base change one shows that the sheaf p∗ωX/B is a line bundleand ωX/B ' p∗(p∗ωX/B). Adopting standard notation, we set ω = R1p∗OX '(p∗ωX/B)∗, where the isomorphism is given by the Grothendieck-Serre duality forp (cf. Eq. (C.12)). Then

ωX/B ' p∗ω∗ . (6.5)

If K = c1(p∗ωX/B) = −c1(ω), the adjunction formula for Θ → X gives

Θ2 = −Θ · p∗K . (6.6)

By [220, Lemma II.4.3], a Weierstraß fibration p admits a Weierstraß form,which one can construct in the following way. Let us consider the projective bundlep : P = P(E∗) = Proj(S•(E))→ B, where

E = p∗OX(3Θ) ∼→ OB ⊕ ω⊗2 ⊕ ω⊗3 .

The divisor 3Θ is relatively very ample and induces a closed immersion of B-schemes j : X → P such that j∗OP (1) = OX(3Θ). The normal sheaf to the localcomplete intersection j, is

NX/P ∼→ p∗ω−⊗6 ⊗OX(9Θ) .

To prove this one takes the Euler exact sequence

0→ ΩP/B → p∗E(−1)→ OP → 0 .

and apply relative duality (Proposition C.1) to the sheaf

ωP/B =∧

ΩP/B ∼→ p∗ω⊗5(−3) .

The morphism p : X → B is a local complete intersection (in the sense of Fulton[119, 6.6]) and has a virtual relative tangent bundle TX/B = [j∗TP/B ] − [NX/P ]in the K-group K•(X). Though TX/B is not a genuine sheaf, it still has Chernclasses; in particular, it has a Todd class which can be readily computed [145].

Proposition 6.11. The Todd class of the virtual tangent bundle TX/B is

td(TX/B) = 1− 12 p∗K +

112

(12Θ · p∗K + 13p∗K2)

− 12

Θ · p∗K2 + terms of higher degree.

If B is smooth, its Todd class is given by the formula

td(B) = 1 +12c1(B) +

112

(c1(B)2 + c2(B)) +124c1(B)c2(B) + . . . .


Thus, we obtain an expression for the Todd class of X:

td(X) = 1 +12p∗(c1(B)− K)

+112

(12Θ · p∗K + 13p∗K2 − 3p∗(c1(B) · K) + p∗(c1(B)2 + c2(B)))

+124

[p∗(c1(B)c2(B))− p∗(K · (c1(B)2 + c2(B))) + 12Θ · p∗(K · c1(B))

+ p∗(c1(B) · K2)− 6Θ · p∗(K2 · c1(B))]

+ terms of higher degree.(6.7)

6.2.2 Torsion-free rank one sheaves on elliptic curves

Let Xt be a fiber of a Weierstraß fibration p : X → B. So, Xt can be any geomet-rically integral Gorenstein curve of arithmetic genus 1 (as we already noted, Xt

has at most one singular point). We denote x0 = σ(t) ∈ Xt; this is a smooth pointin Xt.

For a torsion-free sheaf L of rank one on Xt, the Riemann-Roch theoremyields

χ(L) = h0(L)− h1(L) .

When Xt is smooth, or when is singular and L is of finite Tor-dimension, we havedegL = χ(L). We then adopt this formula as the definition of the degree of atorsion-free sheaf L of rank one on any fiber Xt. Moreover, since the dualizingsheaf of Xt is trivial, the Grothendieck-Serre duality (C.10) implies that

H1(Xt,L)∗ ' HomXt(L,O) = H0(Xt,L∗) , (6.8)

where we write O = OXt for simplicity.

Lemma 6.12. Let L be a torsion-free sheaf of rank one and degree zero on Xt.Then, H0(Xt,L) = H1(Xt,L) = 0 unless L ' O.

Proof. If H0(Xt,L) 6= 0 there is an exact sequence

0→ O → L → K → 0 .

It follows that K has rank zero and length `(K) = χ(K) = χ(L) − χ(O) = 0.Therefore, K = 0 and L is trivial. Moreover, if h0(Xt,L) = 0, then h1(Xt,L) = 0by Riemann-Roch.


Lemma 6.13. Let L be a rank one torsion-free sheaf on Xt. There is a point x ∈ Xt

and an isomorphismL ' mx ⊗O((d+ 1)x0) ,

where d = degL and mx is the ideal sheaf of x in Xt.

Proof. Assume d = −1. In this case, h0(L) = 0 and h1(L) = 1. By Equation (6.8),there is a nonzero morphism L → O which is injective because L is torsion-free.Thus, there is an exact sequence

0→ L → O → K → 0 .

The sheaf K is a quotient of O of rank zero and length `(K) = χ(K) = χ(O) −χ(L) = 1, so that it is the skyscraper sheaf of a point, K ' Ox. Thus, L ' mx. Inthe general case, L⊗O(−(d+1)x0) has degree −1, so that L⊗O(−(d+1)x0) ' mx

by the previous argument.

6.2.3 Relative integral functors for Weierstraß fibrations

Let us consider the commutative diagram

X ×B X

π1

π2 //

ρ

$$HHHHHHHHH X

p

X

p // B

. (6.9)

Definition 6.14. Let I∆ be the ideal sheaf of the diagonal immersion δ : X →X ×B X. The relative Poincare sheaf for the elliptic fibration p is the sheaf

P = I∆ ⊗ π∗1OX(Θ)⊗ π∗2OX(Θ)⊗ ρ∗ω−1

4

We shall show in Section 6.2.4 that P is a universal sheaf for a moduli prob-lem. The restrictions of P to the fibers of either π1 or π2 are torsion-free sheavesof rank one. Moreover, we have twisted the ideal sheaf of the diagonal so as toensure that P satisfies the normalization condition

P|Θ×BX ' P|X×BΘ ' OX . (6.10)

.

Proposition 6.15. The relative Poincare sheaf P has the following properties:


1. it is flat over both factors of X ×B X;

2. its dual sheaf coincides with its dual in the derived category D(X ×B X),i.e., P∗ ' P∨;

3. P∗ is flat over both factors;

4. it is reflexive, i.e., P ' P∗∗.

Proof. 1. It follows from the definition of P.

2. One has to check that ExtiOX×BX (P,OX×BX) = 0 for i ≥ 1. This is a localissue, so by Definition 6.14 it is enough to show that ExtiOX×BX (I∆,OX×BX) = 0for i ≥ 1. Let us consider the exact sequence

0→ I∆ → OX×BX → δ∗OX → 0

where δ : X → X ×B X is the diagonal immersion. By dualizing we obtain anexact sequence

0→ OX×BX → I∗∆ → Ext1OX×BX

(δ∗OX ,OX×BX)→ 0 (6.11)

and isomorphisms

ExtiOX×BX (I∆,OX×BX) ' Exti+1OX×BX

(δ∗OX ,OX×BX) for i ≥ 1.

These sheaves are the cohomology sheaves of the derived homomorphism complexRHomOX×BX (δ∗OX ,OX×BX), which, by Equation (C.7), is isomorphic to thedirect image under δ of the dualizing complex δ!OX×BX . Since π1 δ = IdX ,Equation (C.6) implies that OX ' δ∗p∗2ωX/B [1]⊗δ!OX×BX , and thus δ!OX×BX 'ω−1X/B [−1]. This proves that ExthOX×BX (δ∗OX ,OX×BX) = 0 for h 6= 1.

3. One has to prove that I∗∆ is flat over each factor. The previous computation alsoyields Ext1OX×BX (δ∗OX ,OX×BX) ' δ∗ω−1

X/B , so that the sequence (6.11) takes theform

0→ OX×BX → I∗∆ → δ∗ω−1X/B → 0 . (6.12)

So, I∗∆ has the required property.

4. We need to prove that I∆ ' I∗∗∆ . By Equation (6.12) there is an exact sequence

0→ I∗∗∆ → OX×BX → Ext1OX×BX

(δ∗ω−1X/B ,OX×BX)→ 0 .

By applying relative duality we get

RHomOX×BX (δ∗ω−1X/B ,OX×BX) ' δ∗RHomOX (ω−1

X/B , δ!OX×BX)

' δ∗RHomOX (ω−1X/B , ω

−1X/B [−1]) ' δ∗OX [−1] .

Hence, Ext1OX×BX (δ∗ω−1X/B ,OX×BX) ' δ∗OX and I∗∗∆ ' I∆.


We consider the relative integral functor

Φ: D−(X)→ D−(X)

defined byΦ(E•) = Rπ2∗(π∗1E• ⊗ P) .

Since p is flat, the morphism π1 is flat as well, so that we do not need to deriveπ∗1 . Moreover, P being flat over both factors of X ×B X (Proposition 6.15), thefunctor Φ can be extended to a functor Φ: D(X)→ D(X), which induces a functorbetween the bounded derived categories Φ: Db(X)→ Db(X). We can also regardΦ as an “absolute” integral functor:

Φ(E•) = Φj∗PX→X(E•) = Rπ2∗(π∗1E• ⊗ j∗P) .

Here, j : X×BX → X×X is the natural immersion and π1, π2 are the projectionsof X ×X onto its factors.

W can prove that the functor Φ = Φj∗PX→X : Db(X)→ Db(X) is an equivalenceof categories. To this end we fix some notation. As in Chapter 1 we denote byjx : X → X ×X the immersion of the fiber π−1

1 (x) = x ×X, whilst x : Xt →X ×B X will be the immersion of the fiber π−1

1 (x) ' Xt. Since P and then j∗Pare flat over both factors, one has that

L∗xP ' ∗xP ' Px , Lj∗xj∗P ' j∗xj∗P ' (j∗P)x

and also(j∗P)x ' jt∗Px ,

where jt : Xt → X is the natural immersion.

Assume now that X is smooth. By Proposition C.1, the base variety B isCohen-Macaulay and

ωX ' p∗ωB ⊗ ωX/B ' p∗(ωB ⊗ ω−1) (6.13)

where the second isomorphism is induced by Equation (6.5).

Lemma 6.16. If X is smooth, the sheaf j∗P is strongly simple over both factors ofX ×X.

Proof. Since by Definition 6.14 P is invariant under the permutation of fac-tors, it is enough to check that j∗P is strongly simple over the first factor. Wealready know that it is flat over X, so we only need to compute the groupsHomi

D(X)((j∗P)x, (j∗P)y) for x, y in X (cf. Definition 1.30). Let us write t = p(x),s = p(y). One has

HomiD(X)((j∗P)x, (j∗P)y) ' Homi

D(X)(jt∗Px, js∗Py)

' HomiD(Xt)(Lj

∗t jt∗Px,Py) .


Thus, HomiD(X)((j∗P)x, (j∗P)y) = 0 for any i if s 6= t. Assume now s = t. We

have two possible cases: either x 6= y or x = y. In the first case, either x or y is asmooth point of Xt. If x is smooth in Xt, then Px is a line bundle, which impliesLj∗t jt∗Px ' j∗t jt∗Px ' Px. So, by Lemma 6.12

HomiD(X)((j∗P)x, (j∗P)y) ' Homi

D(Xt)(Px,Py) ' Hi(Xt,P∗x ⊗ Py) = 0

for every i.

Assume on the other hand that x is not smooth in Xt. Since X is smooth, ithas a Serre functor, and we have

HomiD(X)(jt∗Px, jt∗Py)∗ ' Homn−i

D(X)(jt∗Py, jt∗Px ⊗ ωX) .

Now, jt∗Px⊗ωX ' jt∗(Px⊗ j∗t ωX) ' jt∗Px because j∗t ωX ' OXt by (6.13). Thus

HomiD(X)(jt∗Px, jt∗Py)∗ ' Homn−i

D(X)(jt∗Py, jt∗Px) = 0

for every i because y is a smooth point of Xt and we can apply the previousargument.

Finally, if x = y, adjunction between inverse and direct images of sheavesimplies

HomX(jt∗Px, jt∗Px) ' HomXt(j∗t jt∗Px,Px)

' HomXt(Px,Px) ' k ,

where the last isomorphism follows from the stability of Px.

Lemma 6.17. (j∗P)∨ ⊗ π∗1ωX [m+ 1] ' j∗(P∗ ⊗ ρ∗ω−1)[1], where m = dimB.

Proof. Since ωX is a line bundle, one has

(j∗P)∨ ⊗ π∗1ωX [m+ 1] ' RHomOX×X (j∗P, π∗1ωX [m+ 1]) .

Now there are isomorphisms

RHomOX×X (j∗P, π∗1ωX [m+ 1]) ' RHomOX×X (j∗P, π!2OX)

' j∗RHomOX×BX (P, j!π!2OX)

' j∗RHomOX×BX (P, π!2OX)

' j∗RHomOX×BX (P, π∗1ωX/B [1])

' j∗(P∨ ⊗ π∗1ωX/B [1]) ' j∗(P∨ ⊗ ρ∗ω−1)[1] ,

where the first isomorphism is relative duality for π2 together with Equation (C.3),the second is relative duality for j, the third is due to Equation (C.5) and the fourthis relative duality for π2 together with Equation (C.3). In this way we conclude,since P∗ ' P∨ by Proposition 6.15.


Theorem 6.18. If X is smooth, the relative integral functor Φ is a Fourier-Mukaitransform. The quasi-inverse of Φ is the relative Fourier-Mukai functor with rel-ative kernel Q[1], where Q is the sheaf P∗ ⊗ ρ∗ω−1 on X ×B X.

Proof. The sheaf j∗P is strongly simple by Lemma 6.16, so that Φ is fully faithfulby Theorem 1.27. Moreover, for every point x ∈ X, one has (j∗P)x ⊗ ωX 'jt∗(Px) ⊗ ωX ' jt∗(Px ⊗ j∗t ωX) ' jt∗(Px) ' (j∗P)x because j∗t ωX is trivial by(6.13). By Proposition 2.56, Φ is an equivalence of categories and, by Lemma 6.17,its quasi-inverse is the functor

Φ(j∗P)∨⊗π∗1ωX [m+1]X→X ' Φj∗QX→X ,

where m = dimB.

Corollary 6.19. The integral functor Φt = ΦPtXt→Xt : Db(Xt)→ Db(Xt) is an equiv-alence for every closed point t ∈ S. Its quasi-inverse is the Fourier-Mukai functorwith kernel P∗t [1].

Proof. Let H = ΦQ[1]X→X be the quasi-inverse of Φ. Then, the unit morphism Id →

H Φ is an isomorphism and one has an isomorphism jt∗G• → (H Φ)(jt∗G•) forevery object G• of Db(Xt). Since (H Φ)(jt∗G•) ' jt∗(Ht Φt)(G•) by Equation(6.3) and jt is a closed immersion, the unit morphism G• → (Ht Φt)(G•) is anisomorphism; this proves that Φt is fully faithful. Since H is also a left adjointto Φ, a similar argument proves that Ht = ΦP

∗t [1]

Xt→Xt is actually a quasi-inverse ofΦt.

We shall denote by Φ : Db(X)→ Db(X) the relative Fourier-Mukai transformwith kernel Q. Thus, Φt = ΦP

∗t

Xt→Xt for any closed point t ∈ B.

Theorem 6.18 implies that if a sheaf F on X is WITi with respect to Φ(i = 0, 1), then F = Φi(F) is WIT1−i with respect to Φ and Φ1−i(F) ' F . Theanalogous statement intertwining Φ and Φ holds true as well.

6.2.4 The compactified relative Jacobian

Let p : X → B a Weierstraß fibration. Altman and Kleiman proved in [3] thatthere is an algebraic variety p : J0(X/B) → B, the so-called Altman-Kleimancompactification of the relative Jacobian, whose points parameterize torsion-free,rank one and degree zero sheaves on the fibers of X → B. It is a straightforwardconsequence of Lemma 6.13 that the natural morphism of B-schemes

X$−→ J0(X/B)

x 7→ mx ⊗OXt(σ(t)) t = p(x)(6.14)


is an isomorphism (recall that mx is the ideal sheaf of the point x in the fiber Xt)

We shall provide a direct proof of the fact that X is a compactification ofthe relative Jacobian without resorting to Altman-Kleiman’s theory. The Poincaresheaf P will turn out to be a universal object.

Let f : T → B be a scheme morphism. We denote by pT : XT = X×B T → T

and fX : XT → X the projections.

Theorem 6.20. Let L be a sheaf on XT , flat over X, whose restriction Lt to anyfiber Xt is a torsion-free sheaf of rank one and degree zero. There exists a uniquemorphism of B-schemes ψL : T → X such that

(1× ψL)∗P ' L⊗ p∗TM

for a line bundleM on T . Here, 1×ψL is the induced morphism X×BT → X×BX.

Proof. By Equation (C.3) the relative dualizing sheaf for pS is f∗XωX/B . Moreover,f∗XωX/B ' p∗Tω

−1T where ωT ' f∗ω (see (6.5)). Cohomology base change implies

that N = R1pT∗(L ⊗ f∗XOX(−Θ)) is a line bundle, and by relative duality

N−1 ' pT∗HomOXT (L ⊗ f∗XOX(−Θ), p∗Tω−1T ) .

Let us consider the natural morphism

p∗TN−1 → HomOXT (L ⊗ f∗XOX(−Θ), p∗Tω−1T ) .

Its restriction to every fiber Xt is nonzero, so that it induces a section of

HomOXT (L ⊗ f∗XOX(−Θ), p∗T (ω−1T ⊗N )) ,

that is, a morphism L ⊗ f∗XOX(−Θ) → p∗T (ω−1T ⊗N ), whose restriction to every

fiber Xt is nonzero. There is an exact sequence

L ⊗ f∗XOX(−Θ)⊗ p∗T (ωT ⊗N−1)g−→ OXT → OY → 0

for some closed subscheme η : Y → XT = X ×B T .The restriction of g to everyfiber Xt is an injective morphism gt : Lt ⊗ OXt(σ(t)) → OXt . Then, OY is flatover T , i.e., the projection pT η : Y → T is flat. Moreover, Lt ⊗OXt(−σ(t)) hasdegree −1, and by Lemma 6.13 Lt⊗OXt(−σ(t)) ' mx for a certain point x ∈ Xt.Therefore, we have an exact sequence

0→ mxgt−→ OXt → OYt → 0

where Yt = Xt ∩Y . So, OYt is the skyscraper sheaf Ox. Since it is flat, pT η is anisomorphism; thus, Y is the graph of a morphism ψL = fX η(pT η)−1 : T → X.


Moreover, the flatness of OY and the injectivity of all the restrictions gt implythat g is injective and then L ⊗ f∗XOX(−Θ) ⊗ p∗T (ωT ⊗ N−1) ' IY is the idealsheaf of Y . Now, since (1× ψL)∗I∆ ' IY , one has

(1× ψL)∗P ' L⊗ p∗T (ψ∗LOX(Θ)⊗N−1) .

The Poincare sheaf P is the unique universal sheaf on X ×BX for the abovemoduli problem verifying the normalization condition P|Θ×BX ' OX imposed in(6.10).

We denote by ι : J0(X/B) ∼→ J0(X/B) the involution defined by taking thedual. Via the identification $ : X ∼→ J0(X/B) the morphism ι defines an involutionof X, that we denote by the same symbol. There is a functorial description of thisisomorphism: by the universality property, the dual P∗ of the relative Poincaresheaf defines a morphism ι = ψP∗ : X → X such that (1× ι)∗P ' P∗⊗ π∗2M for aline bundle on X. The normalization condition implies that M is trivial, so that

(1× ι)∗P ' P∗ . (6.15)

Remark 6.21. Whenever a fiber Xt is smooth, the fiber J0(X/B)t of the compacti-fication of the relative Jacobian is the dual elliptic curve Xt (which is isomorphic toXt). Moreover, the restriction Pt of the relative Poincare bundle sheaf to Xt× Xt

coincides with the Poincare line bundle defined in Chapter 3. 4

6.2.5 Examples

Assume that X is smooth, so that we can apply Theorem 6.18. We compute theaction of the Fourier-Mukai transform on relatively torsion-free sheaves.

Lemma 6.22. Let L be a torsion-free rank one and degree 0 sheaf on a fiber Xt ofthe Weierstraß fibration p : X → B. Then the direct image Lt = jt∗L is WIT1 forΦ, and Φ1(Lt) ' Ox∗ , where x∗ = ι(x) is the point corresponding to L∗ by theisomorphism $.

Proof. By the invertibility of Φ it is enough to prove that Ox∗ is WIT0 for Φ andthat Φ0(Ox∗) ' Lt. We know that Φ(Oι(x)) ' Lj∗ι(x)∗j∗Q ' j∗Qι(x) because the

kernel Q = P∗ ⊗ ρ∗ω−1 of Φ is flat over the first factor. Since Qι(x) ' P∗ι(x) 'Px ' L, one has Φ(Oι(x)) ' Lt.

Let f : T → B a scheme morphism. Let L be a sheaf on XT flat over Xand whose restrictions Lt to any fiber Xt are torsion-free sheaves of rank one anddegree zero. Let ψL : T → X be the morphism induced by the universal property


(Theorem 6.20), so that (1 × ψL)∗P ' L ⊗ p∗TM for a certain line bundle M onT . The morphism ψL∗ : T → X corresponding to the dual sheaf by the universalproperty is ψL∗ = ι ψL. Thus,

(1× ψL∗)∗P∗ ' L⊗ p∗TM . (6.16)

We denote by Γ: T → XT = X ×B T the graph of ψL∗ , which is a section ofthe projection pT : XT → T . By base change, Γ induces an immersion Γ : XT →XT ×T XT ' (X ×B X)T which is a section of the projection π2T . One also has

1× ψL∗ = fX×BX Γ

where fX×BX : (X ×B X)T → X ×B X is the natural projection.

Proposition 6.23. L is WIT1 for ΦT , and Φ1T (L) ' OΓ(T ) ⊗ p∗T (ωT ⊗M−1).

Proof. By the invertibility of ΦT it is enough to prove that OΓ(T ) is WIT0 for ΦTand that Φ0

T (OΓ(T )) ' L⊗ p∗T (ω−1T ⊗M). Since

π∗1TOΓ(T ) ⊗ P∗T ' Oπ−11T (Γ(T )) ⊗ f

∗X×BXP

∗ ' Γ∗(Γ∗f∗X×BXP∗)

' Γ∗((1× ψL∗)∗P∗) ' Γ∗(L ⊗ p∗TM)

by (6.16), we have

ΦT (OΓ(T )) ' Rπ2T∗(Γ∗(L ⊗ p∗TM))⊗ p∗Tω−1T simeqL ⊗ p∗T (M⊗ ω−1

T ) .

Example 6.24. Take T = B, f = Id and L = OX . The associated morphismψOX : B → X is nothing but the section σ and Γ coincides with σ as well. Moreover(1× σ)∗P = P|X×BΘ ' OX by the normalization condition (6.10). Then

Φ0(OX) = 0 , Φ1(OX) = OΘ ⊗ p∗ω .

4Example 6.25. Take T = X and f = p : X → B. The morphism associated withL = P∗ is ι, and the one associated with P is the identity. The section Γ is thediagonal δ : X → X×BX in the first case, and the composition δ = (1×ι)δ : X →X ×B X in the second case. Hence,

Φ0X(P) = 0 , Φ1

X(P) = δ∗(OX)⊗ ρ∗ω ,Φ0X(P∗) = 0 , Φ1

X(P∗) = δ∗(OX)⊗ ρ∗ω .

4


6.2.6 Topological invariants

In this section we assume that X is smooth. So, every object in Db(X) has a welldefined Chern character. We can compute the Chern character of the complexΦ(E•) by using the Riemann-Roch theorem for π2 since p : X → B is a localcomplete intersection morphism, as we have seen in Section 6.2.1. It should beclear that also π2 is a local complete intersection morphism, whose virtual relativetangent bundle is π∗1TX/B , where TX/B is the virtual relative tangent bundle to pdescribed in Section 6.2.1.

By the singular Riemann-Roch theorem [119, Cor. 18.3.1] the Chern characterof Φ(E•) is

ch(Φ(E•)) = π2∗(π∗1(ch E•) · ch(P) · π∗1 td(TX/B)) . (6.17)

The Todd class of TX/B is given by Proposition 6.11, while the Chern char-acter of P can be computed from Definition 6.14. One has

ch(P) = ch(I∆) · (1 + π∗1Θ +12π∗1Θ2 + . . . ) · (1 + π∗2Θ +

12π∗2Θ2 + . . . )

· (1− ρ∗K +12ρ∗K2 − . . . ) .

(6.18)

Therefore we need to compute the Chern character ch(I∆).

Lemma 6.26. The Chern character of the ideal sheaf I∆ of the diagonal immersionδ : X → X ×B X is

ch(I∆) =1−∆− 12∆ρ∗K + ∆π∗2(Θp∗K) + 5

6∆ρ∗K2

+ 12∆π∗2(Θ p∗K2) + 23

24∆ρ∗K3 + terms of higher degree

where ∆ = δ∗(1) is the class of the diagonal.

Proof. Note first that ch(I∆) = 1− ch(δ∗OX). The singular Riemann-Roch theo-rem gives

ch(δ∗OX)π∗1 td(TX/B) = δ∗(ch(OX)) = δ∗(1) = ∆ .

By using the expression for td(TX/B) given by Proposition 6.11, one has

ch(δ∗OX) =∆ + 12∆ ρ∗K −∆ π∗2(Θ · p∗K)− 5

6∆ ρ∗K2

− 12∆ π∗2(Θ p∗K2)− 23

24∆ ρ∗K3 + terms of higher degree.


Computation for elliptic surfaces

In this case the scheme B is a smooth projective curve and X a smooth projectivesurface. Let us denote by e the degree of the line bundle p∗ωX/B ; recall thatK = c1(p∗ωX/B). We have Θ · p∗K = e = −Θ2 and c1(ωX/B) = p∗K ≡ e f. TheTodd class of the virtual relative tangent bundle of p (Proposition 6.11) is givenby the formula

td(TX/B) = 1− 12 p∗K + ew , (6.19)

where w is the fundamental class of X. The Todd class of X is

td(X) = 1 + 12 (c1 − e)f + ew , (6.20)

where c1 = c1(B). Finally, by Lemma 6.26, the Chern Character of I∆ is

ch(I∆) = 1−∆− 12δ∗(p∗K) + e δ∗(w) .

If E• is an object of Db(X), the Chern character of Φ(E•) is given by

ch(Φ(E•)) =π2∗[π∗1(ch E•) · (1− δ∗(1)− 12 δ∗(p

∗K) + e δ∗(w))ew)

· (1− 12p∗K + ew)] · (1 + Θ− 1

2w) · (1 + e f) .

Thus, the topological invariants of Φ(E•) are

ch0(Φ(E•)) = d

ch1(Φ(E•)) = −c1(E•) + d p∗K + (d− n)Θ + (c− 12 ed+ s) f

ch2(Φ(E•)) = (−c− de+ 12 ne)w ,

(6.21)

where n = ch0(E•), d = c1(E•) · f is the relative degree, c = c1(E•) · Θ andch2(E•) = sw.

Similar calculations for the inverse relative Fourier-Mukai transform yield theformulas

ch0(Φ(E•)) = d

ch1(Φ(E•)) = (c1(E•))− np∗K − (d+ n)Θ + (s+ ne− c− 12 ed)f

ch2(Φ(E•)) = −(c+ de+ 12 ne)w .

(6.22)

Computation for elliptic Calabi-Yau threefolds

We now consider the case of a Weiertraß elliptic fibration p : X → B, where B is asmooth projective surface and X is a projective Calabi-Yau threefold. In this case,the existence of a section of p imposes constraints on the base surface B: it has


to be del Pezzo surface (a surface whose anticanonical divisor −KB is ample), ora Hirzebruch surface (a rational ruled surface), or an Enriques surface (a minimalsurface B for which pa(B) = χ(OB)− 1 = 0, pg = h0(B,KB) = 0 and 2KB = 0)or a blowup of a Hirzebruch surface (see for instance [95] or [223]).

Since ωX ' OX , we have ω ' ωB . By Proposition 6.11, one has

td(TX/B) = 1− 12c1 + 1

12 (13 c21 + 12 Θ c1)− 12Θ c21 (6.23)

with c1 = p∗c1(B) = −p∗(KB). The Todd class of X admits the following expres-sion:

td(X) = 1 + 112p∗(c2 + 11 c21 + 12 Θ c1) (6.24)

with c2 = p∗(c2(B)). Finally, the Chern character of the ideal sheaf I∆ (Lemma6.26) takes the form

ch(I∆) = 1−∆− 12∆ · π∗2c1 + ∆ · π∗2(Θ c1) + 5

6∆ · π∗2(c21) + 12∆ · π∗2(Θ c21) .

We shall consider for simplicity objects E• in D(X) with Chern charactershave the form

ch0(E•) = nE•

ch1(E•) = xE•Θ + p∗SE•

ch2(E•) = Θp∗ηE• + aE• f

ch3(E•) = sE•

(6.25)

where ηE• , SE• ∈ A1(B)⊗Z Q, sE• ∈ A3(X)⊗Z Q ' Q and f ∈ A2(X)⊗Z Q is theclass of a fiber of p. These assumptions are met in the majority of applications.Now Equation (6.17) and the corresponding formula for the inverse elliptic relativeFourier-Mukai transform Φ give the Chern character of Φ(E•) and Φ(E•), namely:

ch0(Φ(E•)) = xE•

ch1(Φ(E•)) = −nE•Θ + p∗ηE• − 12xE•c1

ch2(Φ(E•)) = ( 12nE•c1 − p

∗SE•)Θ + (sE• − 12p∗ηE•c1Θ + 1

12xE•c21Θ)f

ch3(Φ(E•)) = − 16nE•Θc

21 − aE• + 1

2Θc1p∗SE•

(6.26)

and

ch0(Φ(E•)) = xE•

ch1(Φ(E•)) = −nE•Θ + p∗ηE• + 12xE•c1

ch2(Φ(E•)) = (− 12nE•c1 − p

∗SE•)Θ + (sE• + 12p∗ηE•c1Θ + 1

12xE•c21Θ)f

ch3(Φ(E•)) = − 16nE•Θc

21 − aE• − 1

2Θc1p∗SE• + xE•Θc21 .

(6.27)


6.3 Relatively minimal elliptic surfaces

In this section, following [28] and [60], we describe a Fourier-Mukai transform forsmooth elliptic surfaces which need not admit a Weierstraß model. In Chapter7, we shall use this transform to study the Fourier-Mukai partners of an ellipticsurface. Unfortunately this transform does not easily extend to higher dimensions,with the exception of a few cases in dimension three (cf. [71, 82]).

Since elliptic surfaces (Definition 6.9) allow for nonintegral and even reduciblefibers, a compactified relative Altman-Kleiman Jacobian as described in Section6.2.4 may fail to exist. To circumvent this problem, following a suggestion byMorrison [222], one considers a moduli space of pure sheaves supported on thefibers in the sense of Simpson (Definition C.2).

Let p : X → B be our relatively minimal elliptic surface. The dualizing sheafof X can be computed as in the following formula:

ωX ' p∗L ⊗OX(∑

(mi − 1)fi) , (6.28)

where mifi are the multiple fibers of p and L is a line bundle on B [22, V.12.3]. Asa consequence of Equation (6.28), if E• is an object of Db(X) whose cohomologysheaves are all supported on a fiber Xt, then E• and E• ⊗ ωX have the sameChern character. Hence, Equations (1.18) and (1.6) imply that for any object F•of Db(X), the equality

χ(E•,F•) = χ(F•, E•) (6.29)

holds true.

Let us fix some notation. For any object E• of Db(X) we write its Cherncharacter in the form

ch(E•) = (r, c, s) ∈ Z⊕A1(X)⊕Q ,

where r is the rank, c is the first Chern class and ch2(E•) = sw with w thefundamental class of X. We denote by λX/B the highest common divisor of therelative degrees d(E•) = c1(E•) · f of the objects E• of Db(X) (cf. Eq. (6.4)).Equivalently, λX/B is the smallest positive number d such that there is a divisorD with d = D · f. Since the divisor D+ βf is effective for β 0 and has the sameintersection with the fiber as D, we can also say that λX/B is the smallest positiverelative degree d = D · f of an effective divisor D in X (a d-multisection).

Let us fix integer numbers r > 0 and d such that d is coprime to rλX/B .We also fix a polarization H in X having relative degree h = H · f such that rhis coprime to d. To prove that such polarizations actually exist, take an arbitrarypolarization H ′ in X. By the very definition of λX/B , the fiber degree h′ = H ′ · f isa multiple of λX/B ; since d is coprime to rλX/B , by adding if necessary a suitable

6.3. Relatively minimal elliptic surfaces 205

multiple of a λX/B-multisection to H ′, we obtain a new polarization H satisfyingour requirements.

By Theorem C.6, there exists a coarse relative moduli scheme q : M(X/B, r, d)→ B which parameterizes S-equivalence classes of sheaves on the fibers of p thatare relatively semistable with respect to H and have relative polarized rank r (Def-inition C.5) and degree d. This means that the Hilbert polynomial of the sheavesEt is P (s) = rhs+ d.

The coprimality condition ensures that all sheaves in M(X/B, r, d) are rela-tively stable and that M(X/B, r, d) is a fine moduli space (cf. Proposition C.7). Tobe more precise: q is a projective morphism; the (closed) points of M(X/B, r, d) arein a one-to-one correspondence with pure sheaves of polarized rank r and degree don the fibers of p that are µS-stable with respect to the polarization induced by H;and there is a universal relative sheaf P on X ×B M(X/B, r, d)→M(X/B, r, d),flat over M(X/B, r, d), such that for every point y ∈M(X/B, r, d) the restrictionPy of P to the fiber Xq(t) × y is the stable sheaf corresponding to y.

Definition 6.27. The compactified relative Jacobian of type (r, d) is the unionJX/B(r, d) of the connected components of M(X/B, r, d) that contain the directimage it∗(E) of a stable locally free sheaf E of rank r and degree d on a genericfiber Xt of p. 4

In Remark 6.33, we shall compare this compactified Jacobian with the Altman-Kleiman compactified relative Jacobian J0(X/B) previously introduced.

We also denote by P the restriction to X × JX/B(r, d) of the universal sheafon X ×B M(X/B, r, d). Again by Theorem C.6 and the coprimality condition,there also exists a projective variety M(X, r, d) which is a fine moduli scheme forpure dimensional sheaves on X with Chern character v = (0, rf, dw) (where w isthe fundamental class of X) and stable with respect to H (cf. Proposition C.7).Let i : X ×B JX/B(r, d) → X × JX/B(r, d) be the natural immersion. The directimage P = i∗P is flat over JX/B(r, d) and for every point y ∈ JX/B(r, d) its fiberPy ' jt∗(Py) (where t = q(y)) is pure-dimensional and stable with respect to H.Moreover, it has Chern character (0, rf, dw). Then P corresponds to a morphism

ε : JX/B(r, d)→M(X, r, d)

from JX/B(r, d) to the “absolute” moduli scheme M(X, r, d).

For any point y ∈ Y , the sheaves Py = it∗(Py) for are special (Definition2.54), namely, they have the property that Py ' Py ⊗ ωX . This can be seen asfollows: if Py is supported on a smooth fiber Xt of p, then

it∗(Py)⊗ ωX ' it∗(Py ⊗ i∗tωX) ' it∗(Py)


because i∗tωX ' OXt by Equation 6.28. For a general y, there is always a morphismPy → Py ⊗ ωX because y 7→ dim HomX(Py, Py ⊗ ωX) is an upper-semicontinuousfunction [141, III.12.8] [136, 7.7.5]. Since Py and Py⊗ωX are stable with the samec1, the above morphism has to be an isomorphism.

The following result is proved in [60].

Proposition 6.28. q : JX/B(r, d) → B is an elliptic fibration and JX/B(r, d) is asmooth surface. Moreover, ε : JX/B(r, d)→M(X, r, d) is an isomorphism.

Proof. Let us write for simplicity Y = JX/B(r, d). Let U ⊆ B be the largest openset such that p : XU = p−1(U) → U is smooth. For every point t ∈ U the fiberq−1(t) is a moduli space of stable sheaves of rank r and degree d on the ellipticcurve Xt, and then it is isomorphic to Xt by [15]. Now, q is dominant, so that itis surjective and flat, B being a smooth curve [141, III.9.7], and hence there is acomponent of Y that dominates B. Any other connected component of Y mustcontain sheaves supported on a smooth fiber, and this is impossible because thefiber q−1(t) is connected for t ∈ U as we have seen. Then Y is connected andelliptically fibered over B, which proves the first statement.

As for the second statement, note that the support of every sheaf E inM(X, r, d) is contained in a fiber, because its Chern character is (0, rf, dw) andthe support of a stable sheaf is connected. Then ε is one-to-one on closed points. Itfollows that M(X, r, d) is a surface. Since v2 = 0, M(X, r, d) is smooth by Propo-sition 2.61 (see also Remark 2.62). Zariski’s main theorem [141, 11.4] implies thatε is an isomorphism, and then Y is also smooth.

The relative universal sheaf P defines a relative integral functor

Φ = Φ ePY→X : D−(Y )→ D−(X) .

As in many other situations, Φ is defined over the whole of the derived category ofY , and maps Db(Y ) to Db(X). By applying again Proposition 2.61 and Remark2.62 we obtain the following result.

Proposition 6.29. The relative integral functor Φ = Φ ePY→X is an equivalence of

categories.

Corollary 6.30. The elliptic surface q : JX/B(r, d)→ B is relatively minimal.

Proof. Let us write Y = JX/B(1, d) as above. If the claim is not true, there is arational curve C with C2 = −1 contained in a fiber Yt = q−1(t). Then KY ·C = −1,so that χ(OC ,OY ) = χ(C,ωY |C) = 0, whereas χ(OY ,OC) = χ(C,OC) = 1. Itfollows that χ(Φ(OC),Φ(OY )) = 0 and χ(Φ(OY ),Φ(OC)) = 1 because Φ is fullyfaithful. Since p : X → B is relatively minimal, this contradicts (6.29) because

6.3. Relatively minimal elliptic surfaces 207

Φ(OC) and Φ(OC)⊗ωY have the same Chern character, all the cohomology sheavesof Φ(OC) being supported on the fiber Xt.

Remark 6.31. The relative moduli scheme JX/B(r, d) depends on the polarizationH used to define relative stability. Recall that the relative degree h of H has to sat-isfy the coprimality condition gcd(d, rh) = 1. However, if J ′X/B(r, d) is the modulidefined as above with respect to another polarization H ′ (with d coprime to rh′),we have seen that there is an isomorphism of schemes JXU/U (r, d) ' J ′XU/U (r, d),where U ⊆ B is the open set where p : XU = p−1(U) → U is smooth. Since(Corollary 6.30) JX/B(r, d) → B and J ′X/B(r, d) → B are relatively minimal, inview of [220, II.1.2] this isomorphism extends to an isomorphism of elliptic surfacesJX/B(r, d) ' J ′X/B(r, d). Indeed, JX/B(r, d) is independent of the polarization H

(as long as d is coprime to rh). 4

The elliptic surface JX/B(d) = JX/B(1, d) is the relative Picard scheme de-fined by Friedman [110].

Proposition 6.32. Let r, d integers with r > 0 and d coprime to rλX/B. There areisomorphisms

JX/B(r, d) ' JX/B(d) ' JX/B(d+ λX/B)

of elliptic surfaces over B. Thus, if d denotes the residue class of d modulo λX/B,there is an isomorphism JX/B(r, d) ' JX/B(d) of elliptic surfaces over B.

Proof. Let U ⊆ B be the smooth locus of p. The restriction of P ′ to XU ×UJXU/U (r, d) is locally free and its determinant is a line bundle parameterizingline bundles of degree d on the fibers of p : XU = p−1(U) → U . This gives anisomorphism JXU/U (r, d)→ JXU/U (d) which extends to an isomorphism of ellipticsurfaces JX/B(r, d) ' JX/B(d) by a similar argument to the one used in Remark6.31. Let us now onsider a λX/B-multisection Θ. After twisting by OX(Θ) oneobtains an isomorphism JX/B(d) ' JX/B(d+ λX/B).

Remark 6.33. We have seen that in order to ensure that Friedman’s relative Picardscheme JX/B(d) = JX/B(1, d) is projective and a fine moduli space, we have toimpose that d is coprime to λX/B and to the relative degree h of the chosenpolarization. In particular, d = 0 forces λX = 1 and h = 1. The first conditionis equivalent to the fact that X → B has a section, thus preventing X → B

from having multiple fibers; the second imposes that there is a polarization H

that intersects every fiber at one point. Since a polarization must meet all theirreducible components of a fiber, this implies that all fibers are irreducible. Thus,the elliptic surface X → B turns out to be, in this case, a Weierstraß surface.Friedman’s relative Picard scheme JX/B(0) is then isomorphic to the Altman-Kleiman compactified relative Jacobian J0(X/B) as defined in Section 6.2.4, and


by Equation (6.14) we have isomorphisms

X ' J0(X/B) ' JX/B(0) .

Thus, in the case of Weierstraß surfaces, the Fourier-Mukai functors defined inthis section are a generalization of those considered in Section 6.2.3 (which have,however, the advantage of existing in arbitrary dimension). 4

If we drop the coprimality condition on the polarization H, strictly µS-semistable sheaves on the fibers with polarized rank 1 and degree d may exist,and the moduli space q : M(X/B, 1, d)→ B can be considered as a “compactifiedrelative Jacobian,” in the sense that it is a compactification of the moduli spaceof relatively µS-stable sheaves of relative polarized rank 1 and degree d.

6.4 Relative moduli spaces for Weierstraß elliptic fibra-

tions

Let p : X → B be a Weierstraß elliptic fibration. We use the relative ellipticFourier-Mukai transform defined in Section 6.2.3 to study relative moduli spaces ofsheaves on X. In this section we write “(semi)stability” to mean “µS-(semi)stabili-ty” (cf. Definition C.4).

We start by computing the effect of the Fourier-Mukai transform on therelative Chern character.

Proposition 6.34. Let E• a complex in Db(X) of relative Chern character (n, d)(cf. Eq. (6.4)). The relative Chern character of the Fourier-Mukai transform Φ(E•)is (d,−n).

Proof. To compute the relative invariants of Φ(E•), we take for S a point t ∈ Bwith smooth fiber Xt and apply the Riemann-Roch theorem with respect to theprojection of Xt ×Xt onto the second factor.

Let F be a rank n sheaf on X flat over B.

Corollary 6.35. If F is WITi and d 6= 0, then µ(F) = −1/µ(F).

Corollary 6.36. If F is WIT0, then d(F) ≥ 0, and d(F) = 0 if and only if F = 0.If F is WIT1, d(F) ≤ 0.

6.4.1 Semistable sheaves on integral genus one curves

Our aim is to characterize the moduli spaces of µ-semistable sheaves on geomet-rically integral curves of arithmetic genus one, that is, on fibers of a Weierstraß

6.4. Relative moduli spaces for Weierstraß elliptic fibrations 209

elliptic fibration p : X → B. We need to assume that X is smooth so that for everyclosed point t ∈ B, the integral functors Φt : Db(Xt) → Db(Xt) defined for everyfiber are equivalences of categories (cf. Corollary 6.19).

When Xt is smooth, it is an elliptic curve, with origin at the point σ(t). Themoduli spaces of µ-semistable sheaves on it are then characterized, as have seen inSection 3.5.1, by using the Abelian Fourier-Mukai transform S, whose kernel wedenote now by Pab to avoid confusion with the kernel Pt of the elliptic Fourier-Mukai transform Φt. Comparing the expressions of Pab and Φt given respectivelyby Equation (3.12) and Definition 6.14, we see that

Pab ' (1× ι)∗Pt ,

where ι : X → X is the involution which maps a point x to the opposite point forthe group law. Thus,

S ' ι∗ Φt , S ' ι∗ Φt .

Most arguments developed in Section 3.5.1 can be carried through, by using Φtand Φt instead of the Abelian Fourier-Mukai transforms S and S. Although theproofs given there depend on the smoothness of the curve, we can modify them sothat most of the results hold true in the singular case as well. It should be noticedthat when Xt is singular, there exist indecomposable sheaves on Xt (even locallyfree) which are not semistable (cf. [78]). Thus Proposition 3.28 does not generalizeto the singular fibers, and we need to slightly change the strategy of Section 3.5.1.

Let us recall that any torsion-free rank-one sheaf on a fiber Xt is of the formL ' Px for a point x ∈ Xt, and that it is WIT1, with transform Φ1

t (Px) = Ox∗ ,where x∗ = ι(x) (see Lemma 6.22).

To prove that µ-semistable sheaves on a fiber Xt of positive degree are WIT0

we need a preliminary result (cf. also [60]).

Lemma 6.37. A coherent sheaf E on Xt is WIT0 if and only if

HomXt(E ,Px) = 0

for every x ∈ Xt.

Proof. Since Px is WIT1 and Φ1t (Px) = Ox∗ , the Parseval formula (Proposition

1.34) implies that

HomXt(E ,Px) ' HomDb(Xt)(Φt(E),Ox∗ [−1]) .

If E is not WIT0, there is a point x ∈ Xt together with a nonzero morphismΦ1t (E) → Ox. This gives rise to a nonzero morphism Φt(E) → Ox[−1] in the

derived category, so that HomXt(E ,Px∗) 6= 0. The converse is straightforward.


We are now ready to generalize Corollary 3.29 to the case of Weierstraßcurves. As previously discussed, this extends results by Friedman, Morgan andWitten [114], since we consider non-locally free (torsion-free) sheaves.

Proposition 6.38. Let E be a torsion-free µ-semistable sheaf of rank n and degreed on Xt.

1. If d > 0, then E is WIT0 with respect to both Φt and Φt and in both cases Eis µ-semistable .

2. If d < 0, then E is WIT1 with respect to both Φt and Φt and in both cases Eis µ-semistable.

3. If d = 0 and E is µ-stable, then E is of rank one. Thus, any µ-semistablesheaf of degree 0 is WIT1 and E is a skyscraper sheaf.

4. A µ-semistable sheaf E of degree d = 0 is S-equivalent to a direct sum ofdegree zero, rank one torsion-free sheaves:

E ∼m⊕i=1

L⊕nii ,m∑i=1

ni = n .

Moreover E is supported at the points x∗1, . . . , x∗m corresponding to thesheaves L∗i under the identification X ' J0(X/B) of (6.14) and E ' E1 ⊕· · · ⊕ Em, where Ei are µ-semistable subsheaves of degree 0 of E, such thatEi ∼ L⊕nii for every i.

Proof. 1. The fact that E is WIT0 follows from Lemma 6.37, since E is µ-semistableof positive degree. An analogous argument proves that E is WIT0 as well withrespect to Φt. The proof of the semistability of E in both cases is postponed untilthe end of the proof of part 2.

2. The spectral sequence Ei,j2 = Φit(Φjt (E)) (cf. (2.35)) gives rise to an exact

sequence0→ E1,0

2 → E → E0,12 → E2,0

2 = 0 .

Moreover, the sheaf E1,02 = Φ1

t (Φ0t (E)) is WIT0, so that it has positive degree

by Corollary 6.36. E is µ-semistable of negative degree, E1,02 must be zero. Thus,

Φ0t (E) ' Φ0

t (Φ1t (Φ

0t (E))) = 0, and therefore E is WIT1.

Let us check that E is torsion-free and µ-semistable. If the torsion subsheafT of E is nonzero, it is WIT0 (cf. Lemma 6.22), and T is a subsheaf of E havingdegree zero by Corollary 6.36. This contradicts the semistability of E . Thus E istorsion-free of degree n = rk(E) > 0 by Proposition 6.34. If E is not µ-semistable,there is a destabilizing sequence

0→ F → E → G → 0 ,


with F µ-semistable and µ(F) > µ(E). The sheaf G is WIT0; moreover, in viewof point 1, F is WIT0 because it is µ-semistable of positive degree. Thus, we havean exact sequence

0→ F → E → G → 0 ,

so that µ(F) ≤ µ(E) because of the semistability of E . By Corollary 6.35 thiscontradicts the inequality µ(F) > µ(E). The proof for Φ is similar.

We now complete the proof of part 1. Then E has positive degree d > 0 andwe have already seen that it is WIT0 with respect to both Φt and Φt. If E = Φ0

t (E)is not µ-semistable, there is a destabilizing sequence

0→ F → E → G → 0 ,

with F µ-semistable and µ(F) > µ(E). The sheaf F is WIT1, so that d(F) ≤ 0 byCorollary 6.36 and one has exact sequences

0→ Φ0t (G)→ F → K → 0 , 0→ K → E → Φ1

t (G)→ 0 .

If d(F) = 0, F has rank zero, so that K is a torsion sheaf. Then K = 0 andΦ0t (G) ' F ; since the Φ0

t (G) is WIT1 and F is WIT0, one has F = 0 and thenF = 0 which is absurd. Thus, d(F) < 0 and F is µ-semistable by part 2. It followsthat µ(K) ≤ µ(F). Moreover µ(K) ≥ µ(E) by the semistability of E , and thenµ(E) ≤ µ(F). Again by Corollary 6.35 this in contradiction with the inequalityµ(F) > µ(E). The proof for Φ is similar.

3. Since E is µ-stable of degree 0, we have HomXt(E ,Pξ) = 0 unless E ' P∗ξ .Lemma 6.37 implies that if E is not of rank one, it is IT0; by Proposition 3.25 thetransform E = Φ0(E) is then a locally free sheaf of rank 0, so that E = 0, andE = 0 by the invertibility of Φ. This proves the first statement. If E is µ-semistableof degree 0, it has a Jordan-Holder filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = E , withquotients Gi = Ei/Ei−1 being µ-stable of degree 0. The sheaves Gi are torsion-freerank one sheaves of degree 0, that is, Gi ' Pξi for a point ξi ∈ X. Since the sheavesPξi are WIT1 and Pξi ' Oι(ξi), we deduce that E is WIT1 and E is a skyscrapersheaf.

4. The argument used in part 3 proves the statement about S-equivalenceand that E is supported at the points x∗1, . . . , x∗m. Then E ' ⊕mi=1Fi where Fiis a skyscraper sheaf of length ni supported at x∗i . One then takes Ei = Φ0(Fi).

Corollary 6.39. A sheaf E on a fiber Xt of zero degree and rank n ≥ 1 is torsion-freeand µ-semistable if and only if it is WIT1.

Proof. If E is WIT1 all its subsheaves are WIT1 as well; then E has neither sub-sheaves supported on dimension zero, nor torsion-free subsheaves of positive de-


gree, so that it is torsion-free and µ-semistable. The converse is part of Proposition6.38.

The result about categories of µ-semistable sheaves for smooth elliptic curvesproved in Section 3.5.1 can now be extended straightforwardly to the case of gen-eral integral curves of genus one. Let us denote by Cohssn,d(Xt) the full subcategoryof the category Coh(Xt) of coherent sheaves on Xt whose objects are µ-semistablesheaves on Xt of rank n and degree d. We also denote by Skyn(Xt) the categoryof skyscraper sheaves on Xt of length n.

By Proposition 6.38, one has the following result.

Proposition 6.40. The Fourier-Mukai transform Φt induces equivalences of cate-gories

Cohssn,d(Xt) ' Cohssd,−n(Xt) , if d > 0

Cohssn,0(Xt) ' Skyn(Xt) .

Consider the functor Ψt = Φδ∗LtXt→Xt: Db(Xt) ∼→ Db(Xt), with Lt = OXt(σ(t));

so, Ψt(E•) ' E• ⊗ Lt. By composing in a suitable way the integral functors Φtand Ψt and proceeding as in the proof of Proposition 3.31, we obtain the followingresults.

Proposition 6.41. For every pair (n, d) of integers (n > 0), there is an integralfunctor Φt : Db(Xt) ∼→ Db(Xt) which induces an equivalence of categories

Cohssn,d(Xt) ' Cohssn,0(Xt) ,

where n = gcd(n, d). The integral functor Φt Φt induces an equivalence of cate-gories

Cohssn,d(Xt) ' Skyn(Xt) .

Corollary 6.42. Let E be a torsion-free sheaf on Xt of rank n and degree d. Thefollowing conditions are equivalent:

1. E is µ-stable;

2. E is simple;

3. E is µ-semistable and gcd(n, d) = 1.

Thus, the integral functors of Proposition 6.41 map µ-stable sheaves to µ-stablesheaves.


6.4.2 Characterization of relative moduli spaces

Let p : X → B a Weierstraß elliptic fibration whose total space X is smooth. Inthis section we prove that the relative integral functor Φ: Db(X)→ Db(X), whichwe know to be a relative Fourier-Mukai transform by Theorem 6.18, preservesthe relative (semi)stability of sheaves. We shall also use this property to computethe relative moduli spaces of µ-semistable sheaves on the fibers of p. Most ofthe material has been taken from [145, 29]; some results are also contained in[113, 114, 110].

We rely on the results about µ-semistable sheaves on integral genus onecurves described in Section 6.4.1. Recall that if E is a sheaf on X flat over B, therestriction Et of E to the fiber Xt is WITi for every closed point t ∈ B if and onlyif E is WITi and E = Φi(E) is flat over B (cf. Corollary 6.3). By using this facttogether with Proposition 6.38 and Corollary 6.42, we obtain directly the followingresult.

Proposition 6.43. Let E be a relatively µ-(semi)stable sheaf on X.

1. If d > 0, then E is WIT0 with respect to both Φ and Φ, and in both cases Eis relatively µ-(semi)stable.

2. If d < 0, then E is WIT1 with respect to both Φ, and Φ, and in both cases Eis relatively µ-(semi)stable.

3. If d = 0 and E is relatively µ-stable, then E is of rank one. Thus, anyrelatively µ-semistable sheaf of degree 0 is WIT1 and E is a flat family ofskyscraper sheaves of length N = rk(E).

Let us denote by Mss(X/B, n, d) the coarse relative moduli space of rankn and degree d µ-semistable sheaves on the fibers of p (thus, a section of thisspace as a fibration on B corresponds to a relatively µ-semistable sheaf on X).In particular, Mss(X/B, 0, n) is the coarse moduli space of skyscraper sheaves oflength n on the fibers of p, or in other words, the moduli space of µ-semistablesheaves having constant Hilbert polynomial P (m) = n.

Proposition 6.44.

1. There is relative Fourier-Mukai functor Φ : Db(X) → Db(X) which inducesan isomorphism of B-schemes

Mss(X/B, n, d) 'Mss(X/B, n, 0) ,

where n = gcd(n, d).


2. The relative Fourier-Mukai transform Φ induces an isomorphism

Mss(X/B,m, 0) 'Mss(X/B, 0,m) .

Thus, the relative integral functor Φ Φ induces an isomorphism of B-schemes

Mss(X/B, n, d) 'Mss(X/B, 0, n) .

Proof. This follows from Proposition 6.41 and Corollary 6.3.

Next we extend Corollary 3.34 to the relative setting. Since any skyscrapersheaf of length n on a fiber Xt of p is S-equivalent to a direct sum ⊕iOnixi (withn =

∑i ni), the closed points of the relative moduli space Mss(X/B, 0, n) are in

a one-to-one correspondence with the closed points of the relative n-symmetricproduct Symn

B . We shall prove that this correspondence is actually induced by analgebraic isomorphism.

Let us consider the relative Hilbert scheme Hilbn(X/B) → B of B-flat sub-schemes of X of relative dimension 0 and length n. There is a Hilbert-Chow mor-phism Hilbn(X/B) → Symn

B X mapping a zero-cycle of length n to the n pointsdefined by the zero-cycle. Contrary to what happens for a smooth curve, thismorphism is not an isomorphism, but is only birational. However, it induces anisomorphism Hilbn(Xsm/B) ' Symn

B Xsm, where Xsm → B is the smooth locusof p : X → B.

Let T → B be a scheme morphism. If E is a sheaf on X ×B T → T defininga T -valued point of Mss(Xsm/B, 0, n), then the modified support Supp0(E) (seeDefinition C.9) is a subscheme of X×T which is flat over T and has degree n, thatis, a T -valued point of the relative Hilbert scheme Hilbn(X/B). Therefore, we havea functor morphism Mss(Xsm/B, 0, n)→ HomB( •,Hilbn(Xsm/B)), inducing analgebraic morphism between the moduli spaces

ζ : Mss(Xsm/B, 0, n)→ Hilbn(Xsm/B) ' SymnB Xsm . (6.30)

Lemma 6.45. Assume that the base scheme B is normal. The relative moduli spaceMss(X/B, 0, n) of skyscraper sheaves of length n is normal as well.

Proof. If B is a point, X is an elliptic curve andMss(X, 0, n) is smooth (cf. Corol-lary 3.34). Assume that dimB ≥ 1. We first note that since the fibers of the mor-phism Mss(Xsm/B, 0, n) → B are smooth and B is normal, Mss(Xsm/B, 0, n)is normal as well. If ξ is a point of Mss(X/B, 0, n) not in Mss(Xsm/B, 0, n),and t = p(ξ), the fiber Xt has to be singular and ξ belongs to the image of theclosed immersion f : Mss(Xt, 0, n − 1) →Mss(Xt, 0, n) given by F 7→ F ⊕ Ox0 ,where x0 is the unique singular point of the fiber Xt. This proves that the di-mension of Mss(Xt, 0, n − 1) → Mss(Xt, 0, n) is smaller than n − 1. Hence,


the dimension of Mss(X/B, 0, n) − Mss(Xsm/B, 0, n) is smaller that n − 1 +dimB′, where B′ → B is the closed integral subscheme of the points t ∈ B

whose fiber Xt is a singular curve. Since dimB′ < dimB, we conclude that thecodimension of Mss(X/B, 0, n) −Mss(Xsm/B, 0, n) is greater than one. Thus,Mss(X/B, 0, n) is regular in codimension one. It remains only to prove that thedepth ofMss(X/B, 0, n) is greater or equal to 2 at every point ξ ofMss(X/B, 0, n)−Mss(Xsm/B, 0, n). Since t = p(ξ) belongs to B′, it is not the generic point ofB, and we need only to show that that Mss(Xt, 0, n) has depth ≥ 1 at ξ. Since ξlies in the image of the closed immersion f , the result is proved by induction onn.

Proposition 6.46. If B is normal, the morphism ζ of Equation (6.30) is an iso-morphism and extends to an isomorphism of B-schemes

ζ : Mss(X/B, 0, n) ∼→ SymnB X .

Proof. Let T → B be a scheme morphism and ψ : T →∏nB X a morphism of B-

schemes, i.e., a family of morphisms ψi : T → X. Denoting by Γi → X ×B T thegraph of ψi, the sheaf E = ⊕iOΓi is flat over T and restricts to a skyscraper sheaf oflength n on every fiber. Thus, there is a morphism

∏nB X →Mss(X/B, 0, n) given

by ψ 7→ E , which is equivariant under the natural action of the symmetric group S b

and, therefore, induces a morphism of B-schemes η : SymnB X →Mss(X/B, 0, n).

By Proposition 6.40, η is one-to-one on closed points. Since Mss(X/B, 0, n) isnormal by Lemma 6.45, η is an isomorphism by Zariski’s main theorem [141,11.4]. Moreover, we can see that the restriction of η to Symn

B Xsm is the inverseof ζ.

Propositions 6.44 and 6.46 enable us to describe the structure of the relativemoduli spaces of semistable-sheaves on the fibers of p : X → B (cf. [145, Theorem2.1] for the case of degree 0 and [29] for the case of nonzero degree).

Corollary 6.47. Assume that the base variety B is normal.

1. The relative Fourier-Mukai transform Φ induces an isomorphism of B-schemes

ζ : Mss(X/B, n, 0) ∼→ SymnB X

for every positive integer n.

2. For every pair (n, d) of integers (n > 0), there is an isomorphism of B-schemes

Mss(X/B, n, d) ' SymnB X ,

where n = gcd(n, d).


We now study the isomorphism ζ in some more detail. Let E be a µ-semistabletorsion-free sheaf on a fiber Xt, which represents a closed point ofMss(X/B, n, 0).By Proposition 6.38, E is S-equivalent to a direct sum E ∼

⊕ri=0(Li⊕ ni. . .⊕Li) of

torsion-free, rank one sheaves of degree zero (n =∑i ni). The isomorphism ζ can

be explicitly described (on close points) as the assignment

Mss(X/B, n, 0)ζ−→ Symn

B X

E 7→ n0x∗0 + · · ·+ nrx

∗r ,

(6.31)

where x∗i is the point of Xt that corresponds to L∗i under the identification$ : X ∼→ J0(X/B) of (6.14).

By sending L to L∗ ⊗ OX(nΘ), one defines an isomorphism Jn(X/B) ∼→J0(X/B).

Corollary 6.48. There is a commutative diagram of B-schemes

Mss(Xsm/B, n, 0) ∼ //

det

SymnB(Xsm)

φn

J0(X/B) Jn(X/B)∼oo

,

where det is the determinant morphism and φn is the Abel morphism of degree n.

Let Ln be a universal line bundle over q : X ×B Jn(X/B) → Jn(X/B). ThePicard sheaf Pn = R1q∗(L−1

n ⊗ ωX/B) is a locally free sheaf of rank n and definesa projective bundle P(P∗n) = ProjS•(Pn). We have a commutative diagram

Mss(XU/U, n, 0) ∼ // _

SymnU XU

∼ // _

P(P∗n|U ) _

Mss(Xsm/B, n, 0) ∼ //

det

SymnB(Xsm)

Abel

dense // P(P∗n)

uukkkkkkkkkkkkkkk

J0(X/B) Jn(X/B)∼oo

where U → B is the open subset supporting the smooth fibers of p : X → B

and XU = p−1(U). The immersions of the symmetric products into the projectivebundles follow from the structure of the Abel morphism (cf. [3] and Proposition3.24). Let us denote by Pn the locally free sheaf on J0(X/B) induced by Pn viathe isomorphism Jn(X/B) ∼→ J0(X/B).

Corollary 6.49. Pn ' (det)∗(OMss(Xsm/B,n,0)(Θn,0)

).

6.5. Spectral covers 217

By using the isomorphism σ∗(Pn) ∼→ (p∗OX(nH))∗, we obtain the followingtheorem, whose proof is given in [114].

Corollary 6.50. Let Mss(Xsm/B, n,OX) (resp. Mss(XU/U, n,OX)) be the sub-scheme of the sheaves in Mss(Xsm/B, n, 0) (resp. Mss(XU/U, n, 0)) with trivialdeterminant. There is a dense immersion of B-schemes Mss(Xsm/B, n,OX) →P(Vn), where Vn = p∗(OX(nH)). Moreover, this morphism induces an isomor-phism of U -schemes Mss(XU/U, n,OX) ' P(Vn|U ).

Part 1 of Corollary 6.47 generalizes [110, Theorem 3.8] and can be consideredas a global version of the results obtained in Section 4 of [114] about the relativemoduli space of locally free sheaves on X → B whose restrictions to the fibershave rank n and trivial determinant. We can get such results also from Corollary6.48 by making use of the standard properties of the Abel morphism.

6.5 Spectral covers

Let p : X → B be a Weierstraß fibration, with X smooth and B normal as inSection 6.4.2. The isomorphism ζ : Mss(Xsm/B, n, 0) ∼→ Symn

B Xsm provided byProposition 6.46 may be considered from a different perspective. Let Ft be a µ-semistable torsion-free sheaf of rank n and degree 0 on a fiber Xt which definesa closed point of Mss(Xsm/B, n, 0), that is, Ft is S-equivalent to a direct sum ofline bundles of degree 0. Then ζ(Ft) is a cycle of length n defined by the modifiedsupport of the only non-vanishing Fourier-Mukai transform Φ1

t (Ft) of Ft (thismodified support is by definition the closed subscheme defined by 0-th Fittingideal of Φ1

t (Ft), cf. Definition C.9). Recall that, by Proposition 6.38, Φ1t (Ft) is

supported on a finite number of points. When Ft moves in a flat family F , themodified support of Φ1

t (Ft) defines a n-covering of the parameter space of thefamily. We shall study this kind of coverings, which we shall call spectral covers,and see how F can be reconstructed out of its “spectral data.”

Before giving the precise definition of spectral cover, we describe some prop-erties relating the WIT1 condition with relative semistability. The following resultis a direct consequence of Proposition 6.5 and Corollary 6.39.

Proposition 6.51. Let F be a sheaf on X, flat over B and of relative degree zero.There exists an open subscheme S(F) ⊆ B which is the largest subscheme of Bfulfilling one of the following equivalent conditions:

1. FS(F) is WIT1 and FS(F) is flat over S(F).

2. The sheaves Ft are WIT1 for every point t ∈ S(F).

3. The sheaves Ft are torsion-free and µ-semistable for every point t ∈ S(F).


We shall call S(F) the relative semistability locus of F .

Corollary 6.52. Let F be a sheaf on X of fiberwise of degree zero and flat over B.If S(F) is dense, then F is WIT1.

Proof. By Proposition 6.51 FS(F) is WIT1; hence, Φ0S(F)S(F) = 0, because S(F)→

B is a flat base change. The sheaf Φ0S(F) is flat over B and vanishes on an open

dense subset, so that it vanishes everywhere. Thus, F is WIT1.

The notion of spectral cover has been introduced by Friedman, Morgan andWitten [112, 113, 114] (cf. also [10, 145]).

Definition 6.53. Let F be a sheaf on X. The spectral cover of F is the modifiedsupport C(F) = Supp0(Φ1(F)) of Φ1(F), i.e., the closed subscheme of X definedby the 0-th Fitting ideal F0(Φ1(F)). 4

Some fundamental properties of spectral covers are readily established.

Lemma 6.54. The restriction of the spectral cover C(F) to a fiber Xt of p is thespectral cover of the restriction Ft of the sheaf to the fiber,

C(F)t = C(F) ∩Xt ' C(Ft) .

Proof. Since Φ1(F)t ' Φ1t (Ft) by Corollary 6.3, the result follows from the base

change property of the modified support (Lemma C.10).

Lemma 6.55. Let F be a 0-degree torsion-free µ-semistable sheaf of rank n ≥ 1on a fiber Xt and let F ∼

⊕ri=0mL

⊕nii be the S-equivalence given by Proposition

6.38, where L0 is the unique rank 1 torsion-free sheaf of degree 0 on Xt which isnot locally free (and then one may have n0 = 0). Then, length(OXt/F0(F)) ≥ n,where equality holds if either n0 = 0 or n0 = 1, that is, if L0 occurs at most once.

Proof. As we see in the proof of Proposition 6.38, E ' ⊕mi=1Fi where Fi is askyscraper sheaf of length ni supported at the point x∗i corresponding to Li underthe isomorphism X ' J0(X/B). Since the formation of the 0-th Fitting ideal ismultiplicative with respect to direct sums of sheaves (cf. Eq. (C.19)), we haveF0(E) =

∏mi=1 F0(Fi). Moreover, the sheaf ideals F0(Fi) are pairwise coprime,

since they correspond to subschemes supported at different points, and then

OXt/F0(E) 'm⊕i=1

OXt/F0(Fi) .

We now consider the skyscraper sheaves Fi. If the point x∗i is smooth (i.e., if Li is aline bundle), then the local ring OXt,x∗i is a principal ideals domain, and then Fi is

6.5. Spectral covers 219

a direct sum of skyscraper sheaves of the form OXt/mri , where mi is the ideal of x∗i

in Xt. Then F0(Fi) = mnii again by Equation (C.19), and length(OXt/F0(Fi) = ni.

If n0 = 0, that is, if the unique non-locally free rank 1 torsion-free sheafof degree 0 L0 does not occur in the S-equivalence class of E , we deduce thatlengthOXt/F0(E) = n.

To conclude, let us assume that n0 ≥ 1. F0 still has filtration whose successivequotients are isomorphic to OXt/m0, so that Equation (C.20), gives F0(F0) ⊆ mn0

0

and then length(OXt/F0(F0)) ≥ length(OXt/mn00 ≥ n0, with equality only if

n0 = 1. The result follows.

Proposition 6.56. If F is relatively torsion-free and µ-semistable of rank n anddegree zero on X → B, the spectral cover C(F) → B is a finite morphism withfibers of length ≥ n and generic fiber of length n. If in addition C(F) does not meetany singular point of the fibers of p, all the fibers of the spectral cover C(F)→ B

have length n.

Proof. By Lemmas 6.54 and Lemma 6.55, the morphism C(F)→ B is finite withfiber of degree ≥ n. The second statement follows from Lemma 6.55 as well.

A useful result is the following.

Proposition 6.57. If the relative semistability locus of the sheaf F is dense, thespectral cover C(F) contains the whole fiber Xt for every point s /∈ S(F).

Proof. By Corollary 6.52, F is WIT1. Since s /∈ S(F), we have a destabilizingsequence

0→ G → Ft → K → 0 ,

where K is a µ-semistable sheaf on Xs of negative degree. By Proposition 6.43, Kis WIT1 and Φ1(K) is torsion-free. Since Φ1

t (Ft) → Φ1t (K) is surjective, C(F)t =

C(Ft) = Xs.

We know thatF = Φ1(F) = i∗L ,

where i : C(F) → X is the immersion of the spectral cover and L is a sheaf onC(F). What can be said about L? A first look at Lemma 6.22 seems to imply thatL has rank one at every point (at least on the fibers where F is µ-semistable).This is indeed what happens, though one has to be careful because the spectralcover can be quite singular. For a precise statement we need Simpson’s notions oftorsion-free sheaf (Definition C.3) and polarized rank (Definition C.5).

Let us consider on X a polarization of the type

H = aΘ + bp∗HB , a > 0 , b > 0 ,


where HB is a polarization on B.

Proposition 6.58. If F is a relatively torsion-free and µ-semistable sheaf on X → B

of relative rank n and degree zero, then the restriction L of F to the spectral coverC(F) is a torsion-free sheaf of polarized rank one. Conversely, given a closedsubscheme i : C → X such that the projection p i : C → B is a finite coveringof degree n, and a torsion-free sheaf L on C of polarized rank one, then the sheafi∗L is WIT0 with respect to Φ, and F = Φ0(i∗L) is a sheaf on X → B relativelytorsion-free and µ-semistable of rank n and degree zero.

Proof. We first prove that L is a torsion-free sheaf on C(F), or equivalently, thatF is a pure sheaf of dimension equal to dimB on X. Let G be a subsheaf ofF supported in dimension strictly smaller than dimB. For every point t ∈ B therestriction of the spectral cover to the fiber Xt is a finite set. Hence, the restrictionof the modified support of G is finite as well, so that Gt is WIT0 with respect to theinverse Fourier-Mukai transform Φt and G is WIT0 with respect to Φ (Corollary6.3). One has that Φ0(G) is a subsheaf of F . Since Gt = 0 for s /∈ p(Supp0(G))and dim p(Supp0(G)) < dimB, the support of Φ0(G) is not the whole of X, thuscontradicting that F is torsion-free. Moreover, by Proposition C.12, the polarizedrank of F|D(F) is one.

To show the converse, we note that, since the restriction of i∗L to the fiberXt is supported in dimension zero for every point t ∈ B, (i∗L)t is WIT0 withrespect to Φt; by Corollary 6.3 G is WIT0 with respect to Φ as well. Moreover,since the polarized rank of L on C is one, C coincides with the modified supportSupp0(i∗L) by Proposition C.12. Hence, by Proposition C.11, [C] = c1(i∗L). Thenthe relative degree is

d(i∗L) = c1(i∗L) · f = C · f = n .

Since rk(i∗L) = 0, by Proposition 6.34 the relative Chern character of F = Φ0(i∗L)is (n, 0). We have only to check that the restriction Ft of F to every fiber Xt isµ-semistable. Since (i∗L)t is supported in dimension zero, it is IT0, and thenFt = Φ0((i∗L)t) by Corollary 6.3. Thus it is WIT1 and then µ-semistable byCorollary 6.39.

6.6 Absolutely stable sheaves on Weierstraß fibrations

Let p : X → B be a Weierstraß elliptic fibration, and let us assume that X issmooth. We wish to apply the relative Fourier-Mukai transform to the study ofmoduli spaces of sheaves on X that are µ-stable with respect to certain kinds ofpolarizations on X. To distinguish them from the moduli spaces of “relatively”

6.6. Absolutely stable sheaves on Weierstraß fibrations 221

semistable sheaves considered in Section 6.4, we have used the terminology “ab-solutely” stable.

6.6.1 Preservation of absolute stability for elliptic surfaces

We shall use the spectral construction to build µ-stable sheaves on an elliptic sur-face X admitting a Weierstraß model. Actually, stable sheaves on spectral coverstransform to µ-stable sheaves on the surface, and in this way one obtains an opensubset of a moduli space of µ-stable sheaves on the surface. To this end we need tostudy the preservation of stability, and for this, we rely on the computation of theChern character of the Fourier-Mukai transforms provided by Equation (6.21).

The elliptic surfaceX is polarized byH = aΘ+bf for suitable positive integersa and b (cf. Section C.2). Let F be a sheaf on X with Chern character (n,∆, s w)with n > 1, and let w be the fundamental class of X. We denote c = ∆ · Θ andd = ∆ · f as in Equation (6.21), and identify the second Chern character sw withthe rational number s. By using the formula (6.20) for the Todd class of X, wehave the following expressions for the the Hilbert polynomial (cf. Eq. (C.14)) andthe Euler characteristic of F :

χ(X,F(mH)) = 12nH

2m2 + (ac+ bd+ 12na(c1 − e))m+ χ(X,F)

χ(X,F) = s+ 12d(c1 − e) + ne .

(6.32)

Assume now that F is flat over B and that its restrictions to the fibers of Xare torsion-free and µ-semistable sheaves of degree d = 0. By Proposition 6.58, theprojection C(F)→ B of the spectral cover is a finite morphism of degree n. SinceB is smooth, C(F) → B is automatically dominant, and hence is flat. We thenknow that F is a torsion-free sheaf of polarized rank one on the spectral coverC(F), as follows from Proposition 6.58.

Since C(F)→ B is finite, the fiber f induces a polarization on C(F). We canthen consider Simpson stability and semistability with respect to f for sheaves onC(F).

Proposition 6.59. For any integer a > 0 there is an integer b0 > 0 such that, forany b > b0, the sheaf F is µ-(semi)stable on X with respect to H = aΘ + bf if andonly if L = F|C(F) is µ-(semi)stable with respect to f as a sheaf on the spectralcover C(F).

Proof. We can assume a = 1. By Equation (6.21), one has

χ(C(F),L(mf) = χ(X, F(mf)) = nm+ c− ne+ 12nc1 . (6.33)

The Simpson slope of F is

µ(F) =c− ne+ 1

2nc1

n.


Let now0→ G → F → K → 0 (6.34)

be an exact sequence. Then G is supported by C(F), so that it is WIT0 and F ′ = Ghas relative degree 0 and is WIT1 by Proposition 6.58. Reasoning as above, theSimpson slope of G is

µ(G) =c′ − n′e+ 1

2n′c1

n′,

where primes denote the topological invariants of F ′. Moreover one has the exactsequence

0→ F ′ → F → K → 0 .

Assume that F is µ-semistable with respect to H = Θ + bf for some positivenumber b. Then, one has

c′

n′≤ c

n,

because F and F ′ have degree zero on fibers. On the other hand, if F is not µ-semistable and (6.34) is a destabilizing sequence, we have µ(G) > µ(F). This isequivalent to

n′c− nc′ > 0 ,

which is a contradiction. The statement about stability can be proved analogously.

For the converse, assume that F is µ-semistable on X with respect to f andthat

0→ F ′ → F → Q→ 0 (6.35)

is an exact sequence. The sheaf F ′ is WIT1 so that d′ ≤ 0 by Corollary 6.36.Assume first that d′ < 0 and fix b0 > 0 such that H0 = Θ + b0f is a polarization.Then the set of the integers nc1(G) ·H0− rk(G)c, where G ranges over all nonzerosubsheaves of F , is bounded; let ρ be its maximum. Thus, n(c′ + bd′) − n′c ≤ρ+n(b−b0)d′ is strictly negative for sufficiently large b. This proves that if d′ < 0,the exact sequence (6.35) does not destabilize F with respect to any b sufficientlylarge.

Then, if the sequence in Equation (6.35) is destabilizing with respect toH = Θ + bf for some sufficiently large positive number b, one has d′ = 0. We canassume that n′ < n and that Q is torsion-free and µ-semistable with respect toH; the destabilizing condition is now

nc′ − n′c > 0 .

Moreover d(Q) = 0. Since Q is torsion-free, for every t ∈ B there is an exactsequence

0→ F ′t → Ft → Qt → 0


so that Qt is µ-semistable of degree 0. Then Q is WIT1 and one has an exactsequence of Fourier-Mukai transforms:

0→ F ′ → F → Q → 0 .

Proceeding as above we see that the µ-semistability of F implies nc′ − n′c ≤ 0,which is a contradiction.

Proposition 6.59 cannot be directly used in the way it is stated to produceisomorphisms between moduli spaces, because for any given a > 0, the polarizationH = aΘ + bf depends on the sheaf F . Our next aim is to prove that we can findb depending only on the topological invariants of F .

Let us consider a vector v = (n,∆, s w) with n a natural number, ∆ a divisor,s a rational number and w the fundamental class of X. We set c2 = 1

2D2 − s and

B(v) = 2nc2 − (n − 1)∆2. If F is a sheaf with Chern character ch(F) = v,the number B(v) = B(F) is usually called the Bogomolov number of F . If F istorsion-free and µ-semistable with respect to some polarization, the Bogomolovinequality B(F) ≥ 0 holds (cf. [44, 155]). We fix a polarization H0 on X (whichcan be assumed to be of the form H0 = Θ + b0f for some b0 > 0), and we writeHt = H0 + tf. By the Nakai-Moishezon criterion (cf. [141, Theorem A.5.1]), thisis a polarization for every integer t > 0. By technical reasons we will also considerthe real divisors Ht for a real number t. These are elements of the real Neron-Severi group NS(X) ⊗Z R. The ample cone is the convex subset of NS(X) ⊗ZR generated by the (integral) ample divisors; its elements are called real ampledivisors. Stability and semistability with respect to real ample divisors are definedin an analogous way to the ordinary case. Let us assume that F is flat over Band that its restrictions to the fibers are torsion-free and µ-semistable sheaves ofdegree d = 0. Consider n > 1. The proof of the following result is similar to thatof [114, Lemma 7.5].

Lemma 6.60. Assume that L = F|C(F) is µ-stable with respect to f as a sheaf onthe spectral cover C(F). If F is not µ-stable with respect to Ht0 , there exists t ≥ t0and a divisor D such that D ·Ht = 0 and

p ≤ D2 < 0 ,

where p = −n2

4 B(F).

Proof. For the sake of simplicity, we write (semi)stable meaning µ-(semi)stable.By Proposition 6.59, F is Ht-stable for t 0. Let t1 be the smallest real numbergreater or equal to t0 such that F is Ht-stable for every t > t1. It follows easilythat F is strictly Ht1-semistable, so that there is an exact sequence

0→ F ′ → F → F ′′ → 0 ,


where F ′ and F ′′ are torsion-free with the same slope as F with respect to Ht1 .Then, if D = n′c1(F ′′)− n′′c1(F ′), where n′ and n′′ are the ranks of F ′ and F ′′,respectively, one has D ·Ht1 = 0. Moreover, D is not numerically trivial, becauseotherwise, F would be strictly Ht-semistable for all t, so that D2 < 0 by theHodge’s index theorem (cf. [141, Theorem A.5.2]).

On the other hand, one has

B(F) =n

n′B(F ′) +

n

n′′B(F ′′)− D2

n′n′′.

Since F ′ and F ′′ are Ht1 -semistable, B(F ′) ≥ 0 and B(F ′′) ≥ 0 by the Bogomolovinequality, and then D2 ≥ −n′n′′B(F). Furthermore, n = n′ + n′′ gives n′n′′ ≤n2/4 and we have the inequality of the statement.

We can now strengthen Proposition 6.59.

Proposition 6.61. For any integer a > 0, there is b0 > 0 depending only on thetopological invariants (n,∆, s w) = ch(F), such that for any b ≥ b0, the sheaf F isµ-stable on X with respect to H = aΘ + bf if and only if L = F|C(F) is µ-stablewith respect to f as a sheaf on the spectral cover C(F).

Proof. We can take a = 1. We need to prove that there exists b0 > 0 dependingonly on (n,∆, s w) = ch(F), such that if L = F|C(F) is µ-stable with respect to f

as a sheaf on the spectral cover C(F), then F is µ-stable with respect to Θ+bf forany b > b0. This is equivalent to the fact that there exists t0 such that for t ≥ t0,F is µ-stable with respect to Ht.

We divide the proof in two parts.

(a) If D is a divisor such that D2 ≥ p = −n2

4 B(v) and α = D · f > 0, thenD ·Ht > 0 for every t ≥ −(H0 · f) p/2.

Let us consider the divisor E = (Ht · f)D − (D · f)Ht. One has E · f = 0and f2 = 0, and then E2 > 0 by the Hodge index theorem; indeed, the divisorE = (H0 · f)E − (H0 · D·) f is not numerically trivial and E · f = 0 so that0 > E2 = (H0 · f)2E2. It follows that

0 > E2 = λ2D2 − 2λβ D ·Ht + α2(H20 + 2λt) ,

where λ = H0 · f > 0. Since α > 0, one has α2 ≥ 1 and one has

2αλ(D ·Ht) > λ2D2 + α2(H20 + 2tλ) ≥ λ2D2 + (H2

0 + 2tλ) > λ2D2 + +2tλ .

Since λ > 0, one hasα (D ·Ht) > λD2 + 2t ≥ 0 .


(b) F is Ht-stable for t ≥ t0 = n2

8 (H0 · f)B(v) = −(H0 · f) p/2. If F is not Ht0-stable, by Lemma 6.60, there exists t1 ≥ t0 such that F is Ht-stable for t > t1 andstrictly µ-semistable with respect to Ht1 . As in the proof of Lemma 6.60, there isan exact sequence

0→ F ′ → F → F ′′ → 0 ,

where F ′ and F ′′ are torsion-free and Ht1-semistable. If D = n′c1(F ′′)−n′′c1(F ′),one has D · Ht1 = 0 and 0 > D2 ≥ p. Using the same notation as in the proofof Proposition 6.59, we have D · f = −nd′. Since D · f ≤ 0 by part a), whereasthe semistability of the restriction of F to the fibers gives d′ ≤ 0, one has d′ = 0.Then D ·Ht1 = 0 is equivalent to nc′ − n′c = 0, which contradicts the stability ofF . Hence F is Ht0-stable, and is also Ht-stable for t ≥ t0.

Proposition 6.61 has an important consequence.

Corollary 6.62. Let i : C → be an integral Cartier divisor such that C → B is flatof degree n. Given integers a > 0 and d, there exists an integer b0 depending onlyon a and d, such that for every b > b0 and every line bundle L on C of degreed, the Fourier-Mukai transform F = Φ0(i∗L) is a locally free sheaf of rank n ofrelative degree zero on X and is µ-stable with respect to B = aΘ + bf.

This enables us to construct µ-stable locally free sheaves on the elliptic sur-face X out of the “spectral data” (C,L), and then prove that certain moduli spacesof µ-stable sheaves on X are not empty.

6.6.2 Characterization of moduli spaces on elliptic surfaces

It is convenient to polarize both the elliptic surface and the spectral covers withthe same polarization H = aΘ + bf. The advantage is that the (semi)stability ofa sheaf L on a spectral cover i : C → X (with respect to the induced polarizationH|C) is equivalent to the (semi)stability of i∗L with respect to H as a sheaf onX. We have then results analogous to Propositions 6.59 and 6.61. Let (n,∆, s w)be a cohomology class, with n ∈ Z, s ∈ Z and ∆ ∈ H2(X,Z). We write c = ∆ ·Θ,d = ∆ · f and χ = s+ 1

2 (c1 − e)d+ ne (where c1 = c1(B) and e = −Θ2).

Let F be a sheaf on X flat over B with ch(F) = (n,∆, s w) and whoserestrictions to the fibers are torsion-free and semistable sheaves of degree d = 0.

Proposition 6.63. Assume that n > 1. If χ(X, F) > 0, for any integer a > 0 thereis b0 > 0 such that for any b > b0 the following conditions are equivalent.

1. The sheaf F is Gieseker-(semi)stable on X with respect to H = aΘ + bf;

2. F is µ-(semi)stable on X with respect to the same polarization;


3. L = F|C(F) is µ-(semi)stable as pure sheaf of dimension 1 on the spectralcover C(F), with respect to the restriction of H to C(F).

Proof. The second and third conditions are trivially equivalent, so that it is enoughto prove the equivalence between the first and the second. We can take a = 1. Werecall from Equations (6.32) and (6.21) that the Hilbert polynomials of F and Fare given by the formulas

χ(X,F(mH)) = 12nH

2m2 + (c+ 12n(c1 − e))m+ χ ;

χ(X, F(mH)) = (nb− χ)m+ χ ,(6.36)

with χ = χ(X,F) = s + ne + 12c(c1 − e) and χ = χ(X, F) = c − ne + 1

2nc1

(c1 = c1(B)). On the other hand, the Simpson slope of F is

µ(F) =χ

nb− χ.

Let now0→ G → F → K → 0 (6.37)

be an exact sequence. Then G is supported by C(F), so that it is WIT0 and F ′ = Ghas relative degree 0 and it is WIT1 by Proposition 6.58. Reasoning as above, theSimpson slope of G ' F ′ is

µ(G) =χ′

n′b− χ′,

where primes denote the topological invariants of F ′. Since spectral cover C(F ′)is contained in C(F), if we choose an integer b1 > 0 such that H0 = Θ + b1f isample, we have 0 < C(F ′) ·H0 ≤ C(F) ·H0, that is:

0 < b1n′ − χ′ ≤ b1n− χ . (6.38)

Here n′ is the rank of F ′, so that there are only a finite number of possible valuesfor n′, and Equation (6.38) implies that there are only a finite number of possiblevalues for χ′ as well. Moreover these values of n′ and χ′ only depend on thetopological invariants of F .

Let us consider the exact sequence of Fourier-Mukai transforms

0→ F ′ → F → K → 0 ,

and assume that F is Gieseker-stable with respect to H = Θ+bf for some positiveb. Since F and F ′ have degree zero on fibers, this is equivalent to saying thateither nc′ − n′c < 0 or one simultaneously has nc′ − n′c = 0 and nχ′ − n′χ < 0.


We now consider

∆(F ′, F) = (nb− χ)χ′ − (nb− χ′)χ= (nc′ − n′c)b+ cχ′ − c′χ+ n′eχ− neχ′ + 1

2 (nc1χ′ − n′c1χ)

= (nc′ − n′c)(b− χ/n) + χnχ′ − n′χ

n.

(6.39)

On the one hand, since there are only a finite number of values for n′χ−nχ′, andthese values depend only on the topological invariants of F , if nc′−n′c < 0, thereis b0 = b0(n,∆, s w) such that for b > b0 one has ∆(F ′, F) < 0, which proves thestability of F . On the other hand, if nc′−n′c = 0 and nχ′−n′χ < 0, the conditionχ > 0 also implies ∆(F ′, F) < 0.

The corresponding semistability statement is proved analogously.

For the converse, assume that F is µ-semistable with respect to Θ + bf forb 0. Proceeding as in the proof of Proposition 6.59, we see that there exists b0depending on the sheaf F such that if

0→ F ′ → F → Q→ 0

is a destabilizing sequence (in the sense of Gieseker) with respect to H = Θ + bf

for b ≥ b0 with n′ < n and Q torsion-free and Gieseker-semistable with respectto, then d′ = 0. Then, the fact that F ′ destabilizes F is equivalent either tonc′ − n′c > 0, or nc′ − n′c = 0 and nχ′ − n′χ > 0.

Again as in the proof of Proposition 6.59, we prove that Q is WIT1 and thatthere is an exact sequence

0→ F ′ → F → Q → 0 .

From the expression (6.39) for ∆(F ′, F) we see as before that since there areonly a finite number of values for n′χ− nχ′, and these values depend only on thetopological invariants of F , if nc′ − n′c > 0, then there is b0 = b0(n,∆, s w) suchthat for b > b0, one has ∆(F ′, F) > 0. Moreover if nc′−n′c = 0 and nχ′−n′χ > 0,the condition χ > 0 gives that ∆(F ′, F) > 0 also in this case. This contradicts thesemistability of F .

The statement about stability can be proved in a completely similar way.

Proposition 6.63 implies that, if we assume that χ(X, F) < 0, absoluteGieseker-stability with respect to aΘ+ bf is preserved by the Fourier-Mukai trans-form for values of b 0 that depend on a and on the Chern character (n,∆, s w).This was proved in a different way in [145]; similar results can be found in[166, 294].


6.6.3 Elliptic Calabi-Yau threefolds

In this section the base B of the elliptic fibration is a smooth projective surface andwe assume that X is a smooth Calabi-Yau threefold. As we mentioned in Section6.2.1, the existence of a section implies that B has to be a surface of one of thefollowing types: a del Pezzo surface, a Hirzebruch surface, an Enriques surface ora blowup of a Hirzebruch surface.

As we did in Section 6.5, we consider on X a polarization of the type

H = aΘ + b p∗HB , a > 0 , b > 0 ,

where HB is a polarization on B. Our aim is to prove that also for elliptic Calabi-Yau threefolds, the (absolute) µ-stability with respect to H of sheaves on X ofdegree zero on the fibers is preserved by the relative Fourier-Mukai transformfor suitable values of a and b. This was first proved by Friedman, Morgan andWitten [113, 114] for sheaves constructed from spectral data (C,L), where C isan irreducible surface in X, of finite degree n over B, and L is a line bundle onC. In this way they produced instances of µ-stable bundles on elliptic Calabi-Yau threefolds, an issue of great interest in string theory and mirror symmetry,especially in constructing compactifications of the heterotic string.

The proof of the preservation of the absolute µ-stability for arbitrary spec-tral covers was obtained in [4]. As for elliptic surfaces, the proof relies on thecomputation of the Chern character of the Fourier-Mukai transforms provided by(6.26).

We have the following expressions for the the Hilbert polynomial (cf. Eq. (C.14))and the Euler characteristic of F :

χ(X,F(mH)) =16nH

3m3 + 12 ch1(F) ·H2m2

+ (ch2(F) ·H + nH · td2(X))m

+ χ(X,F)

χ(X,F) = ch3(F) + nH · td2(X) ,

(6.40)

where n = ch0(F), c1 = p∗c1(B) and c2 = p∗c2(B). Moreover, by Equation (6.24)the second Todd class of the Calabi-Yau threefold X is td2(X) = 1

12 (c2 + 11c21 +12Θ · c1).

In the following, we shall assume that there is a decomposition

H2i(X,Q) = Θp∗H2i−2(B,Q)⊕ p∗H2i(B,Q). (6.41)

Let us consider a torsion-free sheaf F on X of rank n and degree zero onfibers and write its Chern characters as ch(F) = (n, p∗S,Θp∗η + af, s), where ηand S are classes in A1(B)⊗Z Q, s ∈ A3(X)⊗Z Q ' Q anf f ∈ A2(X)⊗Z Q is theclass of a fiber of p, in agreement with Equation (6.25).


Assume that F is flat over B and that its restrictions to the fibers aresemistable. Then, F is WIT1, the spectral cover C(F) is finite of degree n over B,and by Proposition 6.58, F is a pure sheaf of dimension 1 of polarized rank oneon C(F).

We know from Equation (6.26) that the topological invariants of F are

ch0(F) = 0

ch1(F) = nΘ− p∗η

ch2(F) = −(12nc1 − p∗S)Θ− (s− 1

2p∗ηc1Θ)f

ch3(F) =16nΘc21 + a− 1

2Θc1p∗S .

As in the case of elliptic surfaces, we polarize the spectral cover C(F) withthe restriction HC(F) = p∗HB |C(F) of the pullback of the polarization we havefixed on the base surface B. Since F is supported in codimension 1, by Equation(C.16) the Simpson invariants r(F) and d(F) with respect to HC(F) are

r(F) = nH2B , d(F) = S ·HS −

12bc1 ·HB (6.42)

(recall that td1(X) = 0 because X is a Calabi-Yau threefold).

Proposition 6.64. For any integer a > 0, there is an integer b0 > 0, such that forany b > b0, the following holds true:

1. if F is µ-semistable on X with respect to H = aΘ+bp∗HB, then L = F|C(F)

is µ-semistable with respect to HC(F) as a pure sheaf of dimension 1 on thespectral cover C(F).

2. If L = F|C(F) is µ-stable with respect to HC(F) as pure sheaf of dimension1 on C(F), then F is µ-stable on X with respect to H = aΘ + bp∗HB.

Proof. We can take a = 1. Since F is supported by the spectral cover, the supportof every subsheaf G of F is contained in C(F) as well. Thus, F is WIT0 with respectto the inverse Fourier-Mukai transform and its transform is a WIT1 subsheaf F ofF . Moreover, F has degree zero on fibers, again by (6.27), so that (6.42) remainstrue, mutatis mutandis, for F .

1. Assume that F is µ-semistable with respect to H = Θ + bp∗HB for b 0and that F is destabilized, as a sheaf on the spectral cover, by a subsheaf G. Then,as we said before, G = F |C(F) for certain subsheaf F of V of degree zero on fibers,and we have

µ( F ) > µ(F) ,


which is equivalent tonS ·HB − nS ·HB > 0 .

On the other hand, the µ-semistability of F with respect to H = Θ + bp∗HB gives

2b(nS ·HB − nS ·HB) + (nS − nS) · c1 ≤ 0 ,

for an arbitrarily large b, which is a contradiction.

2. Assume now that L = F|C(F) is µ-stable with respect to HC(F) as a sheafon the spectral cover, and that

0→ F → F → Q → 0

is a destabilizing sequence with respect to H = Θ + bp∗HB . We can assume thatn < n and that Q is torsion-free and µ-stable with respect to H. Moreover, wehave

nc1(F) ·H2 > nc1(F) ·H2 .

The sheaf F is WIT1, so that d ≤ 0 by Corollary 6.36. Assume first that d < 0 andfix b0 > 0 such that H0 = Θ+b0p∗HB is a polarization. Then the set of the integers(nc1(G)−n(G)c1(F))·H2

0 for all nonzero subsheaves G of F is bounded from above;let ρ be its maximum. If Y → X is a surface in the linear system p∗HB , the setof integers (nc1(K)−n(K)c1(F|Y )) ·H0|Y for all nonzero subsheaves K of F|Y hasa lower bound. Hence, the set of the integers (nc1(G) − n(G)c1(F)) · H0 · p∗HB

for all nonzero subsheaves G of F is bounded from above; let ρ′ be its maximum.Now, for b ≥ b0, one has

(nc1(F)− nc1(F)) ·H2 ≤ ρ+ 2(b− b0)ρ′ − ndb2H2B ,

which is strictly negative for b sufficiently large, which is a contradiction. Thend = 0 and the destabilizing condition is

(nS − nS) · (2bHB − c1) ≥ 0 .

One has (nS − nS) · (2b0HB − c1) ≤ ρ′ as before, and then one has

0 ≤ (nS − nS) · (2bHB − c1) ≤ ρ′ + 2(b− b0)(nS − nS) ·HB . (6.43)

Moreover, since Q is torsion-free, for every t ∈ B there is an exact sequence

0→ Ft → Ft → Qt → 0 ,

so that Qt is µ-semistable of degree 0. Then Q is WIT1 and one has an exactsequence of Fourier-Mukai transforms

0→ F → F → Q → 0 .

Proceeding as above we see that the µ-stability of F for b 0 implies that(nS − nS) ·HB < 0. Then the right-hand side of the inequality (6.43) is strictlynegative for b 0, which is a contradiction.


As in the case of elliptic surfaces, one can use Proposition 6.64 to constructµ-stable locally free sheaves on the elliptic Calabi-Yau threefold X out of the“spectral data” (C,L), and to prove that certain moduli spaces of µ-stable sheaveson X are not empty.

Corollary 6.65. Let i : C → X be an integral Cartier divisor such that C → B isflat of degree n. Given integers a > 0 and d, there exists an integer b0 such thatfor every b > b0 and every line bundle L on C of degree d, the Fourier-Mukaitransform F = Φ0(i∗L) is a locally free sheaf of rank n of relative degree zero onX and is µ-stable with respect to B = aΘ + b p∗HB.

The integer b depends on the spectral data (C,L). It is not known whetherit can be chosen as a function only of the topological invariants of (C,L) as it isthe case for elliptic surfaces.


Relative integral functors on singular fibrations. Many of the results stated inthis chapter for elliptic fibrations X → B when the total space X is smooth,still hold true if the smoothness condition is weakened. In particular, the relativeintegral functors defined in Section 6.2.3 are equivalences of categories for arbitraryWeierstraß fibrations. One can prove this statement using the generalization toGorenstein varieties of the characterization of fully faithful integral functors interms of the kernel (see [144]); another proof using spherical objects can be foundin [80].

Compactified Jacobians. Though the compactified relative Jacobians introducedin Section 6.2.4 can be defined for elliptic fibrations X → B of any dimension,their geometry is not known in the general situation. A particularly interestingcase is that of the compactified relative Jacobian J0(X/B) = M(X/B, 1, 0) pa-rameterizing S-equivalence classes of relatively semistable sheaves on the fibershaving relative rank 1 and relative degree 0. One finds out that J0(X/B) → B

is independent of the polarization and is an elliptic fibration as well. Moreover, ifX → B has reducible fibers (hence J0(X/B) contains strictly semistable pointsas pointed out at the end of Section 6.3), the elliptic surface J0(X/B) → B mayeven be singular. In particular, it may fail to be isomorphic to : X → B (cf. [82]).A fairly complete investigation of a class of compactified relative Jacobians, in-cluding those associated to relatively minimal elliptic surfaces, has been carriedout by Lopez Martın in [198, 199].

Stable sheaves on Calabi-Yau threefolds and string theory. The preservation of theabsolute stability provides a method for constructing stable locally free sheaves onminimal elliptic surfaces or elliptic Calabi-Yau threefolds (cf. Propositions 6.62 and


6.65). This method, known as “spectral data construction,” is due to Friedman,Morgan and Witten [113, 114]. This construction of stable bundles on Calabi-Yauthreefolds is of great importance in string theory, since a stable bundle F , satisfyingsuitable properties, provides a compactification of the heterotic string. From aphysical viewpoint, the Calabi-Yau condition and the holomorphicity of the bundlefollow from supersymmetry in the 10-dimensional space-time, and one has also tospecify a B-field. Cancellation of anomalies forces the structure group of the bundleto be E8×E8, SO(32) or one of their subgroups. Because of all these constraints,the bundle F has to satisfy the topological condition ch2(F) = ch2(X) + W ,where W is an effective algebraic class. Moreover, the Euler characteristic χ(F)must be small (physically, the Euler characteristic corresponds to the number ofgenerations of fermionic particles). This appears to be a quite strong restriction.Actually, when X is the quintic in P4 and F is the tangent bundle, one hasχ(F) = 100; similarly, the Euler characteristic is quite big in other examples.

Spectral covers. This construction has received many applications in physics. Anextensive bibliography can be found in the review paper [5].

Gerby transforms. One can also study a kind of transform associated with ellipticfibrations X → B which do not admit a section. To deal with this case, oneneeds to specify some additional data (a B-field from the physical viewpoint, anO∗X -gerbe on X in mathematical terms). This “gerby” transform establishes acorrespondence between some bundle data on X and spectral data on a gerbeassociated with the compactified relative Jacobian. This is treated by Donagi andPantev in [97] and also relates to work of Caldararu [82, 83].

Fourier-Mukai transform for real families of tori. The homological mirror symme-try conjecture by Kontsevich (see, e.g., [188]) postulates (very loosely speaking)an equivalence between the derived category of coherent sheaves on a Calabi-Yaumanifold X, and the so-called Fukaya category of the mirror Calabi-Yau mani-fold X. In a celebrated paper, Strominger, Yau and Zaslow conjectured that anyCalabi-Yau 3-manifold which admits a mirror partner contains an open, dense sub-set which is a fibration in real 3-tori (the SYZ conjecture). Substantial work towarda proof of a (possibily modified) SYZ conjecture has been made by M. Gross [132].Building on this, it has been conjectured (see, e.g., Fukaya in [118]) that homo-logical mirror symmetry is described by a “real” Fourier-Mukai transform. Sucha transform was introduced in [74, 75], see also Arinkin and Polishchuk [8] andLeung, Yau and Zaslow [195]; assuming that X is a symplectic manifold fibered inLagrangian (real) tori, the fiberwise dual fibration X has a natural complex struc-ture, and a suitably defined transform maps local systems supported on Lagrangiansubmanifolds of X to coherent sheaves on X, and vice versa. The construction ofthis transform was further developed in [123].

Chapter 7

Fourier-Mukai partners and

birational geometry

Introduction

In this chapter we offer some applications of Fourier-Mukai transforms, namely, aclassification of the Fourier-Mukai partners of complex projective surfaces, someissues in birational geometry, and an approach to the McKay correspondence viaFourier-Mukai transform.

We have already seen some facts about Fourier-Mukai partners. Two pro-jective varieties X and Y are Fourier-Mukai partners if there is an exact equiva-lence of triangulated categories between their bounded derived categories (Defini-tion 2.36). By Orlov’s representability theorem 2.15, when X and Y are smooththey are Fourier-Mukai partners if and only if there is a Fourier-Mukai functorΦK•

X→Y : Db(X) ∼→ Db(Y ) (Lemma 2.37). As a consequence of this fundamental re-sult, in Chapter 2 we were able to prove that a Fourier-Mukai partner of a smoothvariety is also smooth (Lemma 2.37) of the same dimension and the canonical bun-dles ωX and ωY have the same order (Theorem 2.38). We also know that if twosmooth projective varieties X and Y are Fourier-Mukai partners, and ωX is eitherample or anti-ample, there is an isomorphism X ' Y (Theorem 2.51), that is, theonly Fourier-Mukai partner of X is X itself. In this chapter we shall complete thispicture by developing a classification of all Fourier-Mukai partners Y of a smoothprojective surface X.

Mukai partners have the same Kodaira dimension, and that K-equivalent Fourier-Mukai partners are isomorphic. We shall provide a characterization of crepant


Section 7.1 contains some introductory material. We shall show that Fourier-


234 Chapter 7. Fourier-Mukai partners and birational geometry

morphisms in terms of derived categories. Again as a preliminary tool for studyingFourier-Mukai partners, in Section 7.2 we study integral functors for quotientvarieties.

After quickly showing that an algebraic curve has no nontrivial Fourier-Mukaipartner, we devote Section 7.4 to the study of Fourier-Mukai partners of algebraicsurfaces. We basically follow Bridgeland and Maciocia [70], but we obtain sig-nificant simplifications by using results by Kawamata [175]. We also include thenonmimimal case. In the case of K3 and Abelian surfaces we stay closer to Orlov’sapproach. We also include the computation of the number of Fourier-Mukai part-ners of a K3 surface.

A question which is naturally related to the classification of Fourier-Mukaipartners is the characterization of the autoequivalences of the derived category ofcoherent sheaves. We shall not deal with this problem, just recalling that Bondaland Orlov [49] gave a first contribution to the solution of this question by showingthat if X is a smooth projective variety such that ωX is either ample or anti-ample,then all autoequivalences of Db(X) are generated by shifts, automorphisms of Xand twisting by line bundles. Moreover, the autoequivalences of Db(X) have beencharacterized by Orlov when X is an Abelian variety [243].

In Section 7.5 we shall approach the second topic of this chapter, i.e., therelationship between derived categories and birational geometry, in particular,Bridgeland’s theorem about bounded derived categories of different crepant res-olutions of singularities. This will allow us to show that birational Calabi-Yauthreefolds are Fourier-Mukai partners. The machinery we introduce will includeperverse sheaves and flops.

Section 7.6 is about the McKay correspondence and basically draws fromBridgeland-King-Reid [68]. Let X be a smooth projective variety, acted on bya finite group G such that the canonical bundle of X/G is locally trivial as aG-equivariant sheaf. Then there is a crepant resolution of X/G whose boundedderived category is equivalent to the G-linearized derived category of X, providedthat some dimensional conditions are satisfied (cf. Theorem 7.76).

7.1 Preliminaries

Before approaching the problem of classifying all Fourier-Mukai partners of smoothcomplex projective surfaces, we state some properties of Fourier-Mukai partnersthat hold true in arbitrary dimension. This complements Section 2.3.1. Specificproperties for surfaces or threefolds will be considered later in this chapter.

Proposition 7.1. Let X and Y be smooth Fourier-Mukai partners. There is anisomorphism H0(X,ωiX) ' H0(Y, ωiY ) for every integer i, so that X and Y have

7.1. Preliminaries 235

the same Kodaira dimension.

Proof. Let n be the dimension of both X and Y (cf. Theorem 2.38). Let K• akernel such that ΦK

•

X→Y is a Fourier-Mukai functor. Since the right adjoint of ΦK•

X→Y

is also a quasi-inverse, and equivalences intertwine the Serre functors (Corollary

1.18), one has SkY ' ΦK•

X→Y SkX ΦK•∨⊗π∗Y ωY [n]

X→Y . We have SkY ' ΦδY ∗ωkY [n]

Y→Y andΦK•

X→Y SkX ΦK•∨⊗π∗Y ωY [n]

X→Y ' ΦW•

Y→Y ; here

W• = ΦM•

X×X→Y×Y (δX∗ωkX [n])

withM• = K•LK•∨⊗π∗Y ωY [n] by Proposition 1.3, and δX and δY are the diagonal

immersions X → X × X and Y → Y × Y , respectively. The uniqueness of thekernel (Theorem 2.25) yields δY ∗ωkY ' W•. Finally, ΦM

•

X×X→Y×Y is an equivalenceby Corollary 2.60, so that

HomD(X×X)(δX∗ωjX , δX∗ω

iX) ' HomD(Y×Y )(δY ∗ω

jY , δY ∗ω

iY ) ,

for any pair of integers i and j. Taking j = 0 one finds that H0(X,ωiX) 'H0(Y, ωiY ) as claimed.

For completeness’ sake we recall some definitions and results about singular-ities and their resolutions. A more detailed exposition may be found in [187, 257].

Definition 7.2. A quasi-projective normal variety X has only canonical singulari-ties if it satisfies the following conditions:

1. the canonical divisor KX is a Q-Cartier divisor (i.e., for some integer r > 0the Weil divisor rKX is a Cartier divisor).

2. If p : Z → X is a resolution of singularities of X and Ei are all the excep-tional prime divisors of p, then rKZ = p∗(rKX) +

∑i aiEi, with ai ≥ 0.

If ai > 0 for all the prime divisors Ei, then we say that X has only terminalsingularities. 4

The minimum of the integers r such that rKX is a Cartier divisor is theindex of X.

Definition 7.3. A morphism f : X → Y of Gorenstein varieties is crepant if f∗ωY 'ωY . 4

A weaker notion, which is useful in dealing with Q-divisors, is the following:


Definition 7.4. Let X and Y be quasi-projective normal varieties whose canonicaldivisors are Q-Cartier. A proper birational map α : X 99K Y is crepant if thereare birational morphisms pX : Z → X and pY : Z → Y such that pY = α pX andp∗XKX is Q-linearly equivalent to p∗YKY . 4

In particular, ifX and Y areK-equivalent smooth projective varieties (cf. Def-inition 2.47), then there exists a crepant birational map α : X 99K Y .

We start by recalling a preliminary result.

Lemma 7.5. [175, Lemma 4.2] Let X, Y be quasi-projective normal varieties withonly terminal singularities, and α : X 99K Y a crepant birational map. Then α isan isomorphism in codimension 1, i.e., there exist closed subvarieties X ′ → X

and Y ′ → Y of codimension at least 2, such that α induces an isomorphismX −X ′ ' Y − Y ′.

Since the total transform of any fundamental point of a birational map hasdimension at least 1 [141, V.5.2], we obtain the following result.

Proposition 7.6. Let X and Y be smooth projective surfaces. Any crepant birationalmap α : X 99K Y is an isomorphism. Then, if two smooth projective surfaces Xand Y are K-equivalent, they are isomorphic.

Crepant morphisms can be characterized in terms of the derived category. Inorder to describe such characterization we introduce some notation.

If f : X → Y is a morphism of algebraic varieties and y is a (closed) pointof Y , we denote by Df−1(y)(X) the full triangulated subcategory of Db(X) ofcomplexes topologically supported on the fiber f−1(y) (cf. Definition A.90). Wenote that Df−1(y)(X) contains the structure sheaves of all the infinitesimal neigh-borhoods m · f−1(y) = X ×Y Spec(OY,y/mm

y ) of the fiber, where my denotes themaximal ideal of the local ring OY,y (the stalk of OY at y).

The formal fiber of f over y is the fiber product

f−1(y) = X ×X FSpec(OY,y) ,

where OY,y = proj limmOY,y/mmy and FSpec stands for its formal spectrum (i.e.,

the formal completion of SpecOY,y at its closed point, cf. [141, §II.9]). The for-mal fiber is a formal scheme and can be viewed as the projective limit of theinfinitesimal neighborhoods m · f−1(y) of the fiber.

Lemma 7.7. Assume that Rf∗OX ' OY . A line bundle L on X is the pullback ofsome line bundle N on Y if and only if its restriction to the formal fiber f−1(y)is trivial for every (closed) point y ∈ Y . Moreover in this case one has N ' f∗L.

7.1. Preliminaries 237

Proof. If L ' f∗N , its restriction to each formal fiber is trivial, because the stalkof N at y is isomorphic to OY,y. For the converse, if the restriction of L to eachformal fiber is trivial, by the theorem on formal functions (cf. [141, Thm. III.11.1]),the completions Rif∗Ly and Rif∗OXy of the stalks at any point y of the higherdirect images of L and OX are isomorphic as OY,y-modules. Since Rf∗OX ' OY ,one has Rif∗L = 0 for i > 0 and N = f∗L is a line bundle. One then has an exactsequence

0→ f∗N η−→ L → Q→ 0 , (7.1)

where η : f∗f∗L → L is the natural map, which is injective because f∗N is aline bundle. By the projection formula one has Rif∗(f∗N ) ' N ⊗ Rif∗OX =0 for i > 0. Moreover, since η is the adjunction map between f∗ and f∗, themorphism f∗η has a left inverse, and then it is surjective, which implies thatf∗Q = 0. Applying again the theorem on formal functions we also see that f∗(Q⊗L−1) = 0 and thus, H0(X, f∗(Q ⊗ L−1) = 0. Let us prove that this impliesQ = 0, thus finishing the proof. First, from the exact sequence (7.1) we get aninjective morphism α : f∗N⊗L−1 → OX . Secondly, H0(X, f∗(Q⊗L−1)) = 0 givesH0(X, f∗N ⊗ L−1) ' H0(X,OX) = k, and then f∗N ⊗ L−1 has a nowhere zerosection. It follows that f∗N ⊗L−1 is a trivial line bundle, and then the immersionα is an isomorphism.

Let f : X → Y be a resolution of singularities of a quasi-projective normalvariety Y (that is, X is smooth and f is birational). We assume that Rf∗OX ' OY(i.e., Y has rational singularities) and that f is proper.

Proposition 7.8. If the triangulated category Df−1(y)(X) has trivial Serre functorfor every (closed) point y ∈ Y , then Y is Gorenstein and f : X → Y is crepant.

Proof. The restriction of the Serre functor SX ofX is a Serre functor forDf−1(y)(X).Then, twisting by ωX is the identity on Df−1(y)(X). It follows that the restrictionof ωX to each infinitesimal neighborhood m · f−1(y) of the fiber is trivial, andthen its restriction to the formal fiber f−1(y) is trivial. By Lemma 7.7, f∗ωX isa line bundle, Rf∗ωX ' f∗ωX , and ωX ' f∗f∗ωX . We may prove at the sametime that Y is Gorenstein and f is crepant, by checking that Rf∗ωX [m], wherem = dimX = dimY , is a dualizing complex for Y .

We denote by ΓX and ΓY the functors of global sections on X and Y respec-tively. Then ΓX ' ΓY f∗. For every object F• in the derived category of Y onehas

HomD(Y )(F•,Rf∗ωX [m]) ' HomD(Y )(Lf∗F•, ωX [m])

' Homk(RΓX(Lf∗F•),k)

' Homk(RΓY (Rf∗(Lf∗F•)),k) .


By the projection formula, Rf∗(Lf∗F•) ' F•L⊗Rf∗OX ' F•, so that

HomD(Y )(F•,Rf∗ωX [m]) ' Homk(RΓY (Rf∗(Lf∗F•)),k) .

This proves that Rf∗ωX [m] is a dualizing complex for Y as claimed.

7.2 Integral functors for quotient varieties

When a variety X carries a free action a finite group G, the derived categoryof the quotient variety X/G is equivalent to the equivariant derived category ofthe original variety (cf. Example 1.38). Moreover, in some cases, Fourier-Mukaifunctors between the derived categories of such two quotients X/G and Y /G liftto equivariant Fourier-Mukai functors between the derived categories of X andY . In this section we discuss these issues, following [69] with some modifications(cf. Remark 7.15), which will be useful in Section 7.4.6 when we characterize theFourier-Mukai partners of the Enriques surfaces.

Let X be a smooth projective variety whose canonical line bundle ωX hasfinite order n, that is ωnX ' OX and n is the first exponent such that this propertyholds. Then AX = OX ⊕ ωX ⊕ · · · ⊕ ωn−1

X has a natural structure of algebra overOX and defines a finite etale covering of degree n

ρX : X = SpecAX → X ,

such that AX ' ρX∗OX . Moreover ωX ' ρ∗XωX is trivial. There is a free actionof the cyclic group G = Zn on X, defined by letting the generator εn of G act asthe twist by ωX on AX . The quotient variety X/G is naturally isomorphic to X.We shall call ρX the canonical cover of X.

By Example 1.38, one has equivalences of categories

Lρ∗X : Db(X) ' DG,b(X) , RρGX∗ : DG,b(X) ' Db(X) .

This characterizes the image of the derived inverse image functor. We can also char-acterize the essential image of the direct image functor ρX∗ : Db(X) → Db(X).To do so, let us denote by Spcl(X) the category whose objects are pairs (F , ϕ)where F is a quasi-coherent sheaf on X and ϕ : ωX ⊗F ' F is an isomorphism; amorphism (F , ϕ) → (F ′, ϕ′) is a morphism f : F → F ′ of OX -modules such thatf ϕ = ϕ′ (1⊗ f). Since F is a special sheaf for any object (F , ϕ) ∈ Spcl(X), weshall denominate Spcl(X) the category of special sheaves on X. One easily checksthat Spcl(X) is an Abelian category. The forgetful functor (F , ϕ) 7→ F inducesa functor Db(Spcl(X))→ Db(Qco(X)), which induces a functor Db

c(Spcl(X))→Db(X) between the corresponding subcategories of complexes with coherent co-homology sheaves. An object of Db(X) is in the image of the above functor if it is

7.2. Integral functors for quotient varieties 239

special, or equivalently, if its cohomology sheaves are all special (cf. Proposition2.55).

If E is a quasi-coherent sheaf on X, F = ρX∗E is a quasi-coherent OX moduleendowed with a structure of module over ρX∗OX ' OX ⊕ ωX ⊕ · · · ⊕ ω

n−1X . Such

a module structure is equivalent to the existence of a morphism ϕ : ωX ⊗F → F ,

which on the other hand is an isomorphism because (1 ⊗ ϕ) · · · (1n−1⊗ · · ·⊗ 1 ⊗

ϕ) : F ' ωnX ⊗ F → ωX ⊗ F is its inverse. Then (F = ρX∗E , ϕ) is an object ofSpcl(X) and any such object defines in a quasi-coherent sheaf E on X and anisomorphism F ' ρX∗E . Moreover, E is coherent if and only if F is coherent. Thefollowing proposition is then straightforwardly checked.

Proposition 7.9. The direct image functor induces a functor

ρX∗ : Qco(X)→ Spcl(X) ,

which is an equivalence of Abelian categories and induces an equivalence of derivedcategories

ρX∗ : Db(X) ' Dbc(Spcl(X)) .

Let Y be another smooth projective variety with canonical line bundle oforder n, ρY : Y → Y its canonical cover, and Φ: Db(X) → Db(Y ) an integralfunctor.

Definition 7.10. An integral functor Φ : Db(X) → Db(Y ) is a lift of Φ if thediagram of functors

Db(X)

ρX∗

Φ // Db(Y )

ρY ∗

Db(X) Φ // Db(Y ) ,

commutes, that is, Φ ρ∗X ' ρ∗Y Φ. 4

Lemma 7.11. Let Φ: Db(X) → Db(X) be an integral functor and Φ : Db(X) →Db(X) is a lift of Φ.

1. If Φ = Id, then Φ ' g∗ for some element g ∈ G. In particular, Φ is aFourier-Mukai transform.

2. If Φ = Id, then Φ is a Fourier-Mukai transform and is isomorphic to thetwisting by ωmX for some m ≤ n− 1.

Proof. 1. Since ρX∗ ' ρX∗ Φ, we have that OρX(x) ' ρX∗(Φ(Ox)) for every pointx ∈ X. Then Φ(Ox) ' Og(x) for a point g(x) ∈ X which is in the fiber ρ−1

X (ρX(x)).


By Corollary 1.12 there is a morphism g : X → X and a line bundle L on X suchthat Φ(E•) ' g∗(E• ⊗ L) for every E• ∈ Db(X). Moreover since f = f g, themoprhism g is the multiplication by an element of G that we still denote byg. Let us prove that L ' OX , which finishes the proof of the first part. Onehas ρX∗OX ' ρX∗(Φ(OX) ' ρX∗g∗L ' ρX∗L and then ρ∗X(ρX∗L) ' On

Xsince

ρ∗XωX ' OX . The natural morphism L → ρ∗X(ρX∗L) ' OnX

is nonzero and inducesa nonzero morphism L → OX . Since ρX∗L∗ ' ρX∗(Φ(L∗) ' ρX∗g∗OX ' ρX∗OX ,we prove in the same way the existence of a nonzero morphism L∗ → OX . ThusL is trivial.

2. If x ∈ X and x ∈ ρ−1X (x), then ρX∗ ' Φ ρX∗ gives Φ(Ox) ' Ox. By

Corollary 1.12, there is a line bundle L on X such that Φ(E•) ' E• ⊗ L for everybounded complex E• in Db(X), which proves that Φ is an equivalence of categories.Moreover, ρX∗OXΦ(ρX∗OX) ' ρX∗OX⊗L. Since ρX∗OX ' OX⊕ωX⊕· · ·⊕ω

n−1X ,

one has L ' ωmX for some m ≤ n− 1.

Lemma 7.12. Let Φ = ΦK•

X→Y : Db(X) → Db(Y ) and Φ = Φ fK•X→Y : Db(X) → Db(Y )

be integral functors. Assume that

(ρX × 1)∗K• ' (1× ρY )∗K• .

Then Φ is a lift of Φ.

Proof. Let us consider the diagram

X × YπX

sshhhhhhhhhhhhhhhhhhhhhhhhhh

1×ρYzzuuuuuuuuu

ρX×1 $$IIIIIIIIIπY

++VVVVVVVVVVVVVVVVVVVVVVVVVV

XρX

???????? X × YρX×1

%%JJJJJJJJJ

pXoo X × Y1×ρY

yyttttttttt

pX // YρY

X X × YπXoo πY // Y

For any object E• in Db(X) one has

Φ(ρX∗E•) ' πY ∗(π∗X(ρX∗E•)L⊗K•)

' πY ∗((ρX × 1)∗p∗XE•)

L⊗K•) ' πY ∗(ρX × 1)∗(p∗XE

•L⊗ (ρX × 1)∗K•)

' πY ∗(ρX × 1)∗(p∗XE•

L⊗ (1× ρY )∗K•)

' πY ∗(ρX × 1)∗(1× ρY )∗((1× ρY )∗p∗XE•

L⊗K•)

' ρY ∗RπY ∗(π∗XE•

L⊗K•) ' ρY ∗Φ(E•) ,

7.2. Integral functors for quotient varieties 241

where we have used several times the projection formula in derived category(Proposition A.83).

Let Y be a Fourier-Mukai partner of X and Φ: Db(X) → Db(Y ) an equiv-alence of triangulated categories. By Theorem 2.38, ωY has also order n, and wecan consider its canonical cover ρY : Y → Y .

Proposition 7.13. There is a lift Φ of Φ. Moreover Φ is an equivalence of categories

Φ : Db(X) ' Db(Y ) ,

and is equivariant under the natural action of G on the derived categories in thefollowing sense: there is an automorphism τ of the group G such that g∗ Φ =Φ τ(g)∗ for every g ∈ G.

Proof. Since Φ = ΦK•

X→Y is an equivalence, the integral functor ΦK•∨

Y→X is an equiva-lence as well by Theorem 2.38. Then, by applying Theorem 2.38 to ΦK

•∨

Y→X , we obtainan isomorphism K•⊗π∗Y ωY ' K•⊗π∗XωX . Let us write nowM• = (ρX ×1)∗K• ∈Db(X × Y ). Since ωX×Y ' p∗

XωX ⊗ p∗Y ωY ' p∗Y ωY where pX and pY are the

projections of X × Y onto its factors, one has

M• ⊗ ωX×Y ' ((ρX × 1)∗K•)⊗ p∗Y ωY ' (ρX × 1)∗(K• ⊗ p∗Y ωY )

' (ρX × 1)∗(K• ⊗ p∗XωX) 'M• .

Thus, M• ' (1 × ρY )∗K• for some object K• ∈ Db(X × Y ) by Proposition 7.9applied to the canonical covering X × Y → X × Y . By Lemma 7.12, the integralfunctor Φ = ΦfK•

X→Y : Db(X)→ Db(Y ) is a lift of Φ.

If Ψ is a quasi-inverse of Φ, proceeding as above we prove that there is a liftΨ′ of Ψ. Thus Ψ′ Φ is a lift of the identity and then Ψ′ Φ ' g∗ for some elementg ∈ G by Lemma 7.11. The integral functor Ψ = g−1

∗ Ψ′ is still a lift of Ψ andverifies that Ψ Φ ' Id. Moreover Φ Ψ also lifts the identity, so that Φ Ψ ' h∗for some h ∈ G again by Lemma 7.11. Since hn = 1, one has Id ' (ΦΨ)n ' ΦΨ,where the latter isomorphism follows from Ψ Φ ' Id. Thus Ψ is a quasi-inverseof Φ, so that Φ is an equivalence.

We now prove that Φ is equivariant. If g is an element of G, g∗ Φ is alift of Φ and then Ψ g∗ Φ ' τ(g)∗ for some τ(g) ∈ G by Lemma 7.11. Thusg∗ Φ ' Φ τ(g)∗ and this defines a morphism of groups τ : G → G. One hasanalogously that $(h)∗ Φ ' Φ h∗ defines a morphism of groups $ : G → G

which is inverse to τ . Hence τ is an isomorphism.

Remark 7.14. Since g∗ ' g−1∗ , Φ is equivariant if and only if there is an automor-

phism τ of the group G such that g∗ Φ = Φ τ(g)∗ for every g ∈ G. 4


Remark 7.15. The definition of a lift given in [69] is stronger than ours becausethere the compatibility Φ ρ∗X ' ρ∗Y Φ is also required. Thus our Lemma 7.11is stronger than the corresponding statement [69, Lemma 4.3]. It seems that [69,Lemma 4.4] may fail to hold true unless the additional condition (ρY × 1)∗K• '(1 × ρX)∗K• is imposed. But if one does so, the proof of the existence of the liftgiven in [69] is not complete. 4

7.3 Fourier-Mukai partners of algebraic curves

It is quite simple to show that the only Fourier-Mukai partner of a smooth pro-jective curve is the curve itself. We give an easy proof which uses general resultsand the structure of moduli spaces of stable sheaves on a elliptic curve, which wedescribed in Section 3.5.1.

Theorem 7.16. A smooth projective curve has no Fourier-Mukai partners but itself.

Proof. Let Y be a Fourier-Mukai partner of a smooth projective curve X of genusg. The canonical line bundle ωX of X is either ample (if g > 1), anti-ample (ifg = 0) or trivial (if g = 1). In the first two cases, X ' Y by Theorem 2.51. LetX be elliptic and take a Fourier-Mukai functor Φ: Db(Y ) → Db(X). If y ∈ Y

is a point, by Proposition 2.35 one has∑i dim Hom1

D(X)(Φi(Oy),Φi(Oy)) ≤ 1,

and then there is a unique value of i for which Φi(Oy) 6= 0, that is, Oy is WITi.The integer i is independent of y because of Proposition 6.5. Then Y is a finemoduli space of simple sheaves over X by Corollary 2.64. If the sheaves Φi(Oy)have torsion, they are skyscraper sheaves of length 1, so that Y ' X in this case.If the sheaves Φi(Oy) are torsion-free, they are stable by Corollary 3.33 so thatY ' X by Corollary 3.34.

7.4 Fourier-Mukai partners of algebraic surfaces

In this section we prove the result expressed by the following theorem. We shallassume that the ground field k is the field C of the complex numbers and allsurfaces are smooth and projective.

Theorem 7.17. A smooth projective surface has a finite number of Fourier-Mukaipartners.

Note that by Lemma 2.37 a Fourier-Mukai partner of a smooth projectivesurface is a smooth projective surface as well. Since Fourier-Mukai partners havethe same Kodaira dimension (Proposition 7.1), in order to count the number ofpartners of projective surfaces it seems natural to adopt a case by case approach

7.4. Fourier-Mukai partners of algebraic surfaces 243

essentially based on the Enriques-Kodaira classification. The treatment we offerfollows Bridgeland and Maciocia [70] with some simplifications due to Kawamata[175]. The ideas taken from [175] allow us to consider also surfaces with (−1)-curves, which are not treated in [70].

Let X be a smooth projective surface. We denote by ai the dimension ofthe rational Chow group Ai(X) ⊗ Q. The number a1 is the Picard number ρ(X)of X. If we assume that X is connected, then a0 = a2 = 1. The topologicalEuler characteristic of X is given by χ(X,Q) =

∑i≥0(−1)ibi(X) where bi(X) =

dimHi(X,Q) are the Betti numbers. For X connected one has b0(X) = 1 and alsob4(X) = 1, b3(X) = b1(X) by Poincare duality. Moreover, the topological Eulercharacteristic of X equals the Euler class of the tangent bundle, that is,

χ(X,Q) = c2(X) . (7.2)

Lemma 7.18. Two smooth projective surfaces X and Y that are Fourier-Mukaipartners have the same Picard number, the same Betti numbers, and then, thesame topological Euler characteristic.

Proof. By Corollary 2.40, the rational Chow groups of X and Y are isomor-phic, A•(X) ⊗ Q ' A•(Y ) ⊗ Q, and there are isomorphisms of Q-vector spacesH•(X,Q) ' H•(Y,Q) and H2•(X,Q) ' H2•(Y,Q). We may assume that X isconnected. Then Y is also connected by Proposition 2.53. Thus,

ρ(X) = dimA•(X)⊗Q− 2 = dimA•(Y )⊗Q = ρ(Y )− 2

andb2(X) = dimH2•(X,Q)− 2 = dimH2•(Y,Q)− 2 = b2(Y ) .

Since dimH•(X,Q) = 2 + b2(X) − 2b1(X), we deduce that b1(X) = b1(Y ) aswell.

A useful criterion to establish whether two smooth projective surfaces X andY that are Fourier-Mukai partners are isomorphic can be deduced from the resultsin Section 2.3.1. Let ΦK

•

Y→X : Db(Y )→ Db(X) be a Fourier-Mukai functor. Proposi-tion 2.48 and Theorem 2.49 suggest the importance of the irreducible componentZY (K•) of the support of the kernel K• introduced in Lemma 2.46. Recall thatpY = πX |ZY (K•) : ZY (K•) → Y is dominant, and if ZY (K•) → ZY (K•) is thenormalization of ZY (K•), then the composition morphism pY : ZY (K•) → Y isdominant as well. Moreover, one has

p∗XωrX ' p∗Y ωrY for some r > 0 ,

where pX = πX |ZY (K•) : ZY (K•) → X and pX : ZY (K•) → Y denote the inducedmorphisms.


Proposition 7.19. One has:

1. If dimZY (K•) = 2, then Y and X are isomorphic.

2. If no multiple of KX is zero (that is, if κ(X) 6= 0), then any special sheaf Ehas rank zero and dimZY (K•) ≤ 3.

Proof. 1. By Proposition 2.48, Y and X are K-equivalent, hence they are isomor-phic by Proposition 7.6.

2. Since E ' E ⊗ ωX , the special sheaf E has r = 0, hence E is supportedon curves or points. Moreover, for every point y ∈ Y , p−1

Y (y) is nonempty andcontained in the support of Lj∗yK• ' ΦK

•

Y→X(Oy), which is special by Proposition2.56. Then all its cohomology sheaves are also special, so that dim p−1

Y (y) ≤ 1. Itfollows that dimZY (K•) ≤ 3, as claimed.

The dimension of the variety ZY (K•) takes any possible value, as the followingexamples show:

• if X = Y and K• = O∆ (so that the corresponding integral functor is theidentity), then ZY (O∆) = ∆ which has dimension 2. We shall see nontrivialexamples in Section 7.4.1, among other places.

• If p : X → B is a relatively minimal elliptic surface, Y = JX/B → B is thecompactified relative Jacobian and Φ = Φ eP

Y→X is the relative Fourier-Mukaifunctor defined in Section 6.3, then P is supported on the fiber productY ×B X, and dimZY (P) = 3.

• If the kernel of the Fourier-Mukai functor is a locally free sheaf P on X×Y ,then dimZY (P) = 4. Examples of this situation, which can only occur whenthe Kodaira dimension of X (and then of Y ) is zero, are the Abelian Fourier-Mukai transform (Chapter 3) or the various Fourier-Mukai transforms for K3surfaces described in Chapter 4.

We recall that a smooth algebraic surface X is called minimal if it con-tains no exceptional curves of the first kind, namely, smooth rational curves withself-intersection equal to −1. Exceptional curves of the first kind are also called(−1)-curves. We shall prove Theorem 7.17 by considering several particular cases.We start with the minimal surfaces of Kodaira dimension 2. The case of Kodairadimension κ = −∞ is covered by Section 7.4.2 for the nonelliptic case, and by Sec-tion 7.4.3 for the elliptic case; in both sections some nonminimal surfaces are alsoconsidered. In the case κ = 0 we treat separately the K3 surfaces (Section 7.4.4),the Abelian surfaces (Section 7.4.5), and the Enriques surfaces (Section 7.4.6); theother cases of minimal surfaces with κ = 0, e.g., hyperelliptic surfaces, are again


covered by Section 7.4.3, which includes also the case of Kodaira dimension 1. Theonly remaining surfaces are the nonminimal ones that are not relatively minimalelliptic; this case is eventually treated in Section 7.4.7.

We shall freely use [141, §V.2] and [22, Ch. VI], where the reader can find allthe results about the classification of surfaces that we use here.

7.4.1 Surfaces of Kodaira dimension 2

Proposition 7.20. Let X be a minimal surface of Kodaira dimension 2. If Y is aFourier-Mukai partner of X, then X ' Y .

Proof. By Theorem 2.49, Y and X are K-equivalent, so that they are isomorphicby Proposition 7.6.

In this case, dimZY (K•) = 2. A slightly different proof of Proposition 7.20may be obtained by noting that a projective surface of Kodaira dimension 2 isminimal if and only if its canonical divisor is nef [111, p. 282]. Indeed, by Theorem2.49, the canonical divisor of Y is nef as well, so that Y is also minimal, and thebirational morphism Y → X is actually an isomorphism.

7.4.2 Surfaces of Kodaira dimension −∞ that are not elliptic

By the classification theory of surfaces, a minimal (algebraic) surface X withκ(X) = −∞ which is not elliptic is either the projective plane P2 or a ruledsurface π : X → C over a smooth projective curve C of genus g.

The geometry of ruled surfaces is studied in [141, §V.2], where the readeris referred for additional information. Recall that a ruled surface π : X → C isisomorphic to a projective bundle P(E∗) = ProjS•(E) → C, where E is a rank 2locally free sheaf on C. The bundle E can be normalized so that H0(C, E) 6= 0but H0(C, E ⊗ L) = 0 for any line bundle L of negative degree [141, V.2.8].If E is normalized in this way, the integer e = −deg(E) is an invariant of thesurface. There is a section of π, whose image we denote by C0, such that OX(C0)is the relative line bundle OX/C(1). Notice that a ruled surface may fail to beminimal. This is important because in Section 7.4.7 we shall need to resort to thecomputation of the Fourier-Mukai partners of a (nonminimal) ruled surface we doin this section.

A ruled surface π : X → C has Picard number 2, and its Neron-Severi groupis generated by C0 and the class f of the fiber; they satisfy the relations

C20 = −e , f2 = 0 , C0 · f = 1 .


The canonical divisor of X and the topological Euler characteristic are

KX = −2C0 + (2g − 2− e)f , c2(X) = 4(1− g) . (7.3)

If F is a special sheaf on X of rank r, one has rKX = 0, so that r = 0 sinceno multiple of KX is zero , as follows from Equation (7.3). In other words, F isa torsion sheaf and we can apply Equation (1.7) to obtain χ(F ,F) = −c1(F)2.Moreover, Ext2

X(F ,F)∗ ' HomX(F ,F) by Serre duality because F is special, andone has

dim Ext1X(F ,F) = 2 dim HomX(F ,F) + c1(F)2 ≥ 2 + c1(F)2 . (7.4)

Lemma 7.21. Let π : X → C be a ruled surface. If D is an irreducible curve in X

such that D ·KX = 0, then D2 ≥ 0 unless g = 0 and e = 2. Moreover, if g = 0and e = 2, then D = C0.

Proof. Since D · KX = 0, if D and KX are linearly dependent over Q, one hasD2 = 0. Otherwise, (KX , D) is a basis for the vector space A1(X)⊗Q. If D2 < 0,the Hodge index theorem implies that K2

X > 0. Thus g = 0, that is, X is rational,and then e ≥ 0. If D = aC0 + bf , one has 0 ≤ D · f = a because D is irreducible.By the same reason, 0 ≤ D ·C0 = b−ae unless D = C0. Since 0 > D2 = a(2b−ae),one cannot have 0 ≤ b− ae, and then D = C0, so that 0 = C0 ·KX = e− 2. Thisproves the first part. The second claim follows easily from [141, V.2.18].

Proposition 7.22. Let X be minimal surface with κ(X) = −∞ that admits noelliptic fibration. The unique Fourier-Mukai partner of X is X itself.

Proof. Let Φ = ΦK•

Y→X : Db(Y ) → Db(X) be a Fourier-Mukai functor. If X ' P2,then ωX is anti-ample, so that X ' Y by Bondal and Orlov’s reconstruction theo-rem (Theorem 2.51). Assume then that X is a ruled surface over a curve C of genusg with invariant e. By Proposition 7.19, dimZY (K•) ≤ 3 and if dimZY (K•) = 2then X ' Y , so that we can assume that dimZY (K•) = 3. This excludes thesituation g = 0 and e = 2: indeed, in this case C0 is the unique irreducible curveD such that D · KX = 0 by Lemma 7.21, and then the support of Φ(Oy) mustconsist of a point for generic y ∈ Y ; this implies dimZY (K•) = 2.

Our first aim is to prove that Oy is WITi for some integer i for every pointy ∈ Y . Indeed Lemma 7.21 implies one has c1(Φi(Oy))2 ≥ 0 for every i, and thenif more than one of the sheaves Φi(Oy) is nonzero, one has∑

i

dim Hom1D(X)(Φ

i(Oy),Φi(Oy)) ≥ 4


by Equation (7.4); this contradicts Proposition 2.35. In principle, the integer imight depend on the point y, but it does not due to Proposition 6.5. Thus, Φ(Oy) 'Φi(Oy)[−i], so that

c1(Φi(Oy))2 = c1(Φ(Oy))2 = −χ(Φ(Oy),Φ(Oy)) = −χ(Oy,Oy) = 0

by the Parseval formula (Proposition 1.34). It follows that all fibers of πY : ZY (K•)→ Y are curves Dy = Supp Φ(Oy) such that D2

y = 0 and Dy · KX = 0. Byadjunction all curves Dy have arithmetic genus 1, so that the smooth ones areelliptic. Moreover, Dy1 ·Dy2 = 0 when y1 6= y2 due to Equation (1.7).

Now, we fix a very ample divisor H on Y with H ·KY 6= 0; we may assumethat H is smooth. If Ψ: Db(X) → Db(Y ) is a quasi-inverse of Φ, for any pointx ∈ X the support of Ψ(Ox) meets H at a finite number of points, because Ψ(Ox)is special. We are going to prove that this induces a morphism X → Symd(H) forsome integer d which is an elliptic fibration onto its image.

Let us consider the integral functor ΨH = Lj∗ Ψ: Db(X)→ Db(H) wherej : H → Y is the immersion. IfQ• is the kernel of Ψ, then ΨH has kernelQ•|X×H =L(1×j)∗Q•. We now prove that Q•|X×H [−1] reduces to a single sheaf which is flatover X. This is equivalent to proving that all the sheavesOx are WIT0 with respectto ΨH [−1]. Since O∨H ' OH(H)[−1], the Parseval isomorphism (Proposition 1.34)gives

HomiDb(X)(L,ΨH(Ox)[−1]) ' Homi

Db(X)(L,OHL⊗Ψ(Ox)[−1])

' HomiDb(Y )(L ⊗O

∨H ,Ψ(Ox)[−1])

' HomiDb(Y )(L ⊗OH(H),Ψ(Ox))

' HomiDb(X)(Φ(L ⊗OH(H)),Ox)

for any line bundle L on Y . If we take L to be very ample, Φ(Ln ⊗ OH(H)) =RπX∗(π∗Y (Ln ⊗OH(H))⊗P) reduces to a locally free sheaf in degree 0 for n bigenough. By Lemma 2.8, ΨH(Ox)[−1] reduces to a single sheaf in degree 0, that is,Ox is WIT0 with respect to ΨH [−1] for every x.

Since the support of any Ψ(Ox) meets H at a finite number of points, wehave proved that the sheaf Q•|X×H [−1] defines a flat family of skyscraper sheaveson H, so that it defines a morphism

f : X → Sd(H)

for some integer d > 0, where Sd(H) is the (coarse) moduli space of skyscrapersheaves on H of length d (cf. Theorem C.6). Given a point ξ ∈ Sd(H) defined bythe equivalence class of the sheaf Om1

y1⊕ · · · ⊕ Omsys , where d =

∑mi, the fiber

f−1(ξ) is the intersection of the curves Dy1 , . . . , Dys . Since Dyi ·Dyj = 0 for i 6= j,


we see that the schematic image of f is isomorphic to the curve H embedded intoSd(H) by y 7→ Ody . Therefore, the generic fiber of f : X → H is an elliptic curve,contradicting our hypothesis.

7.4.3 Relatively minimal elliptic surfaces

We prove now that the partners of relatively minimal elliptic surfaces (cf. Defini-tion 6.9) are precisely the relative compactified Jacobians JX/B(d) = JX/B(1, d)introduced in Section 6.3. Our treatment in this section includes all surfaces withκ = 1, the only case with κ = −∞ that is still to be covered (i.e., the elliptic case),and some cases with κ = 0.

Proposition 7.23. Let p : X → B be a relatively minimal elliptic surface (Definition6.8). Any Fourier-Mukai partner Y of X is isomorphic to the relative compactifiedJacobian JX/B(d) for some integer d coprime to λX/B.

Proof. Recall that by Equation (6.28), KX is a rational multiple of the ellipticfiber f. Then, the support of any nonzero special sheaf is contained in a finitenumber of fibers of p.

Now, let Y be a Fourier-Mukai partner of X and Φ = ΦK•

Y→X a Fourier- Mukaifunctor. Take a (closed) point x in a smooth fiber of p and a point y ∈ Y suchthat the support of Φ(Oy) contains x; they exist by Propositions A.91 and 2.52.Since HomD(X)(Φ(Oy),Φ(Oy)) = C, the support of Φ(Oy) is connected; the objectΦ(Oy) being special, the support is either the point x or the fiber p−1(p(x)). Inthe first case, Φ(Oy) ' Ox[i] for some integer i, and Proposition 6.6 implies thatX and Y are K-equivalent; by Proposition 7.6, X and Y are isomorphic. Thuswe assume that the support of Φ(Oy) is the fiber p−1(p(x)), so that the Cherncharacter of Φ(Oy) is (0, rf, d) for some integers r > 0 and d. By Equation (1.4)and Parseval formula (Proposition 1.34), one has

1 = χ(Oy,OY ) = χ(Φ(Oy),Φ(OY )) = rc1 · f− c0d

where c0 and c1 are the zeroth and first Chern character of Φ(OY ). Since c1 · fis an integer multiple of λX/B by definition of λX/B , one has that rλX/B is co-prime to d. Then, the relative compactified Jacobian JX/B(r, d) (Definition 6.27)exists and is a smooth surface equipped with a relatively minimal elliptic fibrationq : JX/B(r, d) 'M(X, r, d)→ B (Proposition 6.28 and Corollary 6.30). We are go-ing to see that actually Y ' JX/B(r, d), which concludes the proof by Proposition6.32.

Since the sheaf Φ(Oy) is supported on a elliptic curve, one has Hom1X(Φi(Oy),

Φi(Oy)) 6= 0 for every i and then Proposition 2.35 implies that actually there isonly one nonzero cohomology sheaf. In other words, the sheaf Oy is WITi for some


i for every y ∈ Y . The integer i is independent of the point y by Proposition 6.5.Furthermore, Φi(Oy) is simple and then it is stable after restriction to its supportsince the latter is an elliptic curve (see Corollary 3.33). Thus, Φi(Oy) defines aclosed point of Y = JX/B(r, d).

By Corollary 2.64, Φ induces an isomorphism between Y and a fine modulispace for the sheaves Φ(Oy). Since JX/B(r, d) is also a fine moduli space, Φ inducesan immersion Y → Y = JX/B(r, d) which has to be an isomorphism because bothvarieties are projective surfaces, and JX/B(r, d) is irreducible since it is an ellipticsurface.

7.4.4 K3 surfaces

By a result of Orlov, the Fourier-Mukai partners of a K3 surface are completelycharacterized in terms of isometries of the transcendental lattice (see Definition4.9).

Theorem 7.24. [242, Thm. 3.3] Let X, Y be two projective K3 surfaces. X andY are Fourier-Mukai partners if and only if the transcendental lattices T(X) andT(Y ) are Hodge isometric.

Proof. Let Ψ = Db(X) → Db(Y ) be an equivalence of categories. By Theorem2.15, there exists a kernel K• such that ΦK

•

X→Y ' Ψ. The proofs of the first assertionin Proposition 4.27 and of Corollary 4.28 apply to this situation, showing that theinduced map f : T(X)→ T(Y ) is a Hodge isometry.

Conversely, assume that a Hodge isometry g : T(Y )→ T(X) is given. Sincethe orthogonal complement of T(X) in H•(X,Z) contains the hyperbolic sublatticeU , by Proposition B.8 the map g extends to a Hodge isometry g : H•(Y,Z) →H•(X,Z). Let v = (g($))∗, where $ is the fundamental class in H4(Y,Z), andwrite v = (r,H, t). We may recast v into a standard form where r > 0 and H isample by using the isometries of H•(X,Z) of the form

v 7→ v · (1, `, 12`

2) with ` ∈ Pic(X) (7.5)

(s, `, z) 7→ (z, `, s) , (7.6)

which restrict to the identity on T(X). Indeed we may note that v cannot be of theform (0, L, 0), so that, possibly by applying the isometry (7.6) and changing signto v, we may assume that r > 0. Moreover, by repeatedly applying the isometry(7.5) with ` an ample class, we may assume that H is ample, and may be taken asa polarization in X. Since $ is primitive and isotropic, so is v. By [227, Thm. 5.4]the moduli space MH(v) of H-stable bundles on X is nonempty. Moreover, thereexists an element u ∈ H1,1(X,Z) such that v · u = 1 (take u = −g(1)). Then byTheorem 4.20 there is a universal family E on X ×MH(v), while by Proposition


4.21 the integral functor ΦQX→MH (v) is a Fourier-Mukai transform, and by Theorem4.25 MH(v) is a K3 surface. Therefore one has a Hodge isometry

f = fQ : H•(X,Z)→ H•(MH(v),Z) .

Now, for any K3 surface S, one has H2(S,Z) ' $⊥/Z$, where $ is the funda-mental class in H4(S,Z). As a consequence, the composition g−1 f−1 establishesa Hodge isometry between H2(MH(v),Z) and H2(Y,Z). By the weak Torelli the-orem (Corollary 4.11), the K3 surfaces MH(v) and Y are isomorphic, so thatDb(X) ' Db(Y ).

Remark 7.25. By Proposition B.8, Theorem 7.24 can be alternatively stated asfollows: the derived categories Db(X) and Db(Y ) are equivalent if and only if thefull Mukai lattices H•(X,Z) and H•(Y,Z) are Hodge isometric. 4

The proof of Theorem 7.24 implies the following remarkable result, whichshows that, to a certain extent, the geometry of the moduli space of stable sheaveson a K3 surface X is encoded into the derived category Db(X). (Again, in the re-mainder of this section by “(semi)stable” we shall mean “Gieseker-(semi)stable.”)

Theorem 7.26. Let X, Y be two projective K3 surfaces. Y is a Fourier-Mukaipartner of X if and only if it is isomorphic to a 2-dimensional fine compact modulispace of stable sheaves on X (and vice versa).

The examples in Section 4.3 realize indeed this situation.

A natural question is whether the number of Fourier-Mukai partners of aprojective K3 surface X is finite or not, and in the former case, to write down acounting formula for it. The first result in this direction was already contained in[227], Proposition 6.2.

Proposition 7.27. If the Picard number of a projective K3 surface X is greaterthan 11, then X has no other Fourier-Mukai partner than itself.

Proof. Let Y be a Fourier-Mukai partner, and let g : T(X) → T(Y ) be a Hodgeisometry, which exists by Theorem 7.24. Thanks to Proposition B.8, g extends toan isometry g : H2(X,Z)→ H2(Y,Z). Hence X and Y are isomorphic by the weakTorelli theorem (Corollary 4.11).

Since the Picard number of a Kummer surface (see Example 4.5) is at least17, this implies that a Kummer surface has no Fourier-Mukai partner other thanitself.

Another simple instance is provided by an elliptic K3 surface X with a sec-tion.


Proposition 7.28. A projective elliptic K3 surface X with a section has no otherFourier-Mukai partner than itself.

Proof. The fiber of the projection X → P1 and the section generate a hyperboliclattice sitting in Pic(X). Then the claim follows from Proposition B.8 by reasoningas in Proposition 7.27.

The finiteness of the number of Fourier-Mukai partners of any K3 surfaceX was first proved by Bridgeland and Maciocia [70]. Here we shall follow [150],where an explicit counting formula is also given. We shall denote by FM(X) theset of nonisomorphic Fourier-Mukai partners of X.

Given a K3 surface X, a primitive embedding i : T(X)→ Σ (where Σ is thestandard K3 lattice) determines a point in the period domain ∆ ⊂ P(Σ⊗C). Noteindeed that T(X) has a natural Hodge structure induced by the Hodge structureof H•(X,Z). The complexification T(X)⊗C contains the line H2,0(X,C), and theimage of this line via i provides the aforementioned point in ∆. By the surjectivityof the period map there is a marked K3 surface (Y, φ) corresponding to this point.Now, the trascendental lattice T(S) of a K3 surface S has the property of be-ing the minimal primitive sublattice of H2(S,Z) whose complexification cointainsH2,0(S,C). This implies the existence of a commutative diagram

T(Y ) _

g // T(X) _

i

H2(Y,Z)

φ // Σ

where g is a Hodge isometry. In view of Theorem 7.24, Y is a Fourier-Mukaipartner of X.

Lemma 7.29. Two primitive embeddings i, i′ : T(X) → Σ determine isomorphicK3 surfaces Y , Y ′ if and only if there is an α ∈ O(Σ) and a Hodge isometry h ofT(X) such that the diagram

T(X) h // _

i

T(X) _

i′

Σ

α // Σ

(7.7)

commutes.

Proof. Follows from the (weak) Torelli theorem, Corollary 4.11.


Let J be the subgroup of O(T(X)) formed by Hodge isometries. For lateruse we provide the following characterization of J ; for a proof see [150, Prop. B1].

Proposition 7.30. J is isomorphic to Z/2mZ for some m ∈ N. Moreover, the orderof the subgroup J× of the invertibles of J divides the rank of T(X).

According to the terminology of Section B.3, we say that two primitive em-beddings i, i′ : T(X) → Σ are J-equivalent if they fit into a diagram such asEquation (7.7).

Corollary 7.31. Let

EJ(T(X),Σ) = i : T(X) → Σ | primitive embedding/J-equivalence .

There is a bijective map

µ : EJ(T(X),Σ)→ FM(X) .

Theorem 7.32. The set FM(X) is finite.

Proof. In view of the previous discussion, in particular Corollary 7.31, the resultfollows from Lemma B.9.

Remark 7.33. This result combined with Theorem 7.26 implies that the numberof nonisomorphic 2-dimensional fine compact components of the moduli space ofstable sheaves on a K3 surface is finite. 4

One can write a formula computing the number of elements in FM(X), whichfollows straightforwardly from Theorem B.10. Let g(Pic(X)) = K1, . . . ,Ks bethe genus of the lattice Pic(X), with K1 ' Pic(X) (for the notion of genus seeSection B.1).

Theorem 7.34. [150] The cardinality of the set FM(X) is given by

](FM(X)) =s∑i=1

](O(Ki)\O(AKi)/J)

where AKi is the discriminant group of the lattice Ki (see Section B.2).

The counting formula takes a particularly elegant form when ρ(X) = 1.

Corollary 7.35. [239, 150] Let X be a projective K3 surface of Picard number 1,with a generator H such that H2 = 2d. Then ](FM(X)) = 2ω(d)−1, where ω(1) = 1and ω(d) is the number of prime factors in d for d ≥ 2.


Proof. One has APic(X) ' Z/2dZ with the quadratic form q(1) = 1/2d. Then byTheorem B.4, we have g(Pic(X)) = Pic(X). Moreover, O(Pic(X)) ' Z/2Z. ByProposition 7.30, the group J is Z/2Z as well. By the counting formula,

](FM(X)) = 12 ](O(Z/2dZ)) .

It is now easy to show that O(Z/2dZ) ' (Z/2Z)ω(d), cf. e.g. [271].

In the case of Picard number ρ(X) equal to 2, the counting of the Fourier-Mukai partners of X appears to be related to issues of a number-theoretic nature.One has for example the following result [150]. We denote by d(Pic(X)) the dis-criminant of the Picard lattice Pic(X).

Proposition 7.36. Let X be a projective K3 surface with ρ(X) = 2 and d(Pic(X)) =−p for some prime number p 6= 2. Then

](FM(X)) = 12 (h(p) + 1)) ,

where h(p) is the class number of the extension Q(√p).

When the Picard number of X is greater than 2, the set FM(X) consists ofjust one element.

Corollary 7.37. If the Picard number of X is at least 3, and the discriminantd(Pic(X)) of Pic(X) is square-free, then X has no other Fourier-Mukai partnerthan itself.

Proof. Since the order of the discriminant group APic(X) equals the absolute valueof d(Pic(X)), the group APic(X) is cyclic, so that l(APic(X)) = 1, where l(G) isthe minimal number of generators of a finite group G. By applying [235, Theorem1.14.2] one sees that

](FM(X)) = ](O(Pic(X))\O(APic(X))) = 1.

7.4.5 Abelian surfaces

Fourier-Mukai partners of Abelian surfaces may be characterized exactly as in thecase of K3 surfaces. Before analyzing this problems, let us notice that we knowfrom Chapter 3 that an Abelian variety X and its dual X are Fourier-Mukaipartners. For an Abelian surface X, the transcendental lattice T(X) is defined asfor K3 surfaces as the orthogonal lattice to Pic(X) in H2(X,Z).


Theorem 7.38. Let X, Y be two Abelian surfaces. X and Y are Fourier-Mukaipartners if and only if the transcendental lattices T(X) and T(Y ) are Hodge iso-metric.

Proof. The proof is the same as for Theorem 7.24 up to the point where we get aHodge isometry f : H2(X,Z)→ H2(Y,Z). In this case, Theorem 1 of [268] impliesthat Y is isomorphic either to X or to the dual Abelian surface X. In either case,we get a Fourier-Mukai partner of X.

Again as in the case of K3 surfaces, one proves the finiteness of the numberof Fourier-Mukai partners of Abelian surfaces.

Corollary 7.39. An Abelian surface has only finitely many Fourier-Mukai partners.

Another characterization of the Fourier-Mukai partners of an Abelian varietywas given by Orlov in [243] (note that this characterization works in any dimension,not just for surfaces, and for every ground field). For the sake of completeness westate it here without proof. LetX, Y be Abelian varieties, and let f : X×X ∼→ Y×Ybe an isomorphism, which we represent in the matrix form

f =(a b

c d

).

We say that f is isometric if the inverse f−1 has the matrix form

f−1 =(d −b−c a

).

Theorem 7.40. [243, Thm. 2.19] Two Abelian varieties X and Y are Fourier-Mukaipartners if and only if there is an isometric isomorphism X × X ∼→ Y × Y .

Orlov used this to prove that an Abelian variety (of arbitrary dimension,defined on any ground field) has only finitely many Fourier-Mukai partners.

7.4.6 Enriques surfaces

Fourier-Mukai partners of Enriques surfaces have been computed by Bridgelandand Maciocia in [70] building on their previous work about Fourier-Mukai trans-form for quotient varieties [69] and on some results by Nikulin about lattices [235],also used in the study of Fourier-Mukai partners of K3 and Abelian surfaces.

We recall that an Enriques surface is a projective smooth minimal surfaceX whose canonical line bundle ωX is of order 2, that is, ω2

X ' OX and ωX is


nontrivial. Then A = OX ⊕ωX has a natural structure of OX -algebra and definesa finite etale covering of degree 2

ρX : X = SpecAX → X ,

such that AX ' ρX∗OX . We call ρX the canonical cover of X. The twist of AXby ωX defines a free action of G = Z/(2) on X, and one has X/G ' X. Moreover,ωX is trivial so that X is a K3 surface. Since ρX is finite and X is a minimalsurface, X is minimal as well.

The generator ε of G acts on the integer cohomology H•(X,Z) and gives riseto a an orthogonal decomposition

H•(X,Z) ' H•+(X,Z) ⊥ H•−(X,Z)

as a direct sum of the sublattices where ε acts as the identity and as the multi-plication by −1, respectively. One has H•−(X,Z) ⊂ H2(X,Z) and H0.2(X,C) ⊂H•−(X,Z)⊗Z C. One also has an orthogonal decomposition

H2(X,Z) ' H2+(X,Z) ⊥ H•−(X,Z)

which proves that H2+(X,Z) is even and unimodular and is the sublattice orthog-

onal to H•−(X,Z) in H2(X,Z). Moreover, the pullback by ρX gives an immersion

ρ∗X : H2(X,Z)/Torsion → H2+(X,Z) .

Since H2(X,Z)/Torsion is indefinite by [22, VIII.15.1], H2+(X,Z) is indefinite as

well. In particular, it is 2-elementary in the sense of [235, Def. 3.6.1 ].

Let Y be a Fourier-Mukai partner of X and Φ: Db(X)→ Db(Y ) an equiva-lence of triangulated categories. By Theorem 2.38, Y is also an Enriques surfaceand there is a canonical cover ρY : Y → Y which identifies Y with the quotient ofa free action of G on a K3 minimal surface Y .

By Proposition 7.13 (cf. also Remark 7.14), there is a lift of Φ, that is, a G-equivariant equivalence of triangulated categories Φ : Db(X) → Db(Y ) such thatε∗ Φ = Φ ε∗ (because the unique automorphism of Za is the identity). Thus,the induced Hodge isometry

f : H•(X,Z)→ H•(X,Z)

(cf. Proposition 4.27) is G-equivariant.

Proposition 7.41. The surface Y is isomorphic to X, that is, an Enriques surfacehas no Fourier-Mukai partners but itself.


Proof. The G-equivariant Hodge isometry f induces a G-equivariant isometryf− : H•−(X,Z)→ H•−(Y ,Z). Let us prove that it extends to an isometry

f : H2(X,Z)→ H2(Y ,Z) .

Since H2(X,Z) and H2(Y ,Z) are isometric, we need to check that any isome-try of H•−(X,Z) is induced by an isometry of H2(X,Z). There is an orthogonaldecomposition H2(X,Z) ' H2

+(X,Z) ⊥ H•−(X,Z), and then the orthogonal toH•−(X,Z) in H2(X,Z) is H2

+(X,Z). As we have already shown, the latter is aneven, indefinite and 2-elementary lattice; then [235, Thm. 3.6.2, Thm. 3.6.3] implythat we are in the hypotheses of [235, Prop. 1.14.1]. This enables us to conclude.

We thus have proved that f− extends to an isometry f : H2(X,Z)→ H2(Y ,Z).Moreover, f is automatically a G-equivariant Hodge isometry. By the Torelli the-orem for Enriques surfaces [22, VIII.21.2], X and Y are isomorphic.

7.4.7 Nonminimal projective surfaces

In this section we complete the study of the Fourier-Mukai partners of a smoothalgebraic surface. The case which still remains open is that of some nonmini-mal surfaces. We have however already proved in Proposition 7.23 that relativelyminimal elliptic surfaces (Definition 6.9) have a finite number of Fourier-Mukaipartners, even if they may fail to be minimal. Here we study the remaining casesof nonminimal surfaces following Kawamata’s treatment [175].

Let us recall that if two smooth projective surfaces X and Y are Fourier-Mukai partners and ΦK

•

X→Y : Db(X) → Db(Y ) is a Fourier-Mukai functor, thereexist an irreducible component Z of the support W of K• (cf. Definition A.90),such that the projection pX = πX |Z : Z → X is surjective. By Lemma 2.46, ifZ → Z is the normalization morphism, and pX : Z → X, pY : Z → Y are theprojections, then pX is dominant and p∗Xω

rX ' p∗Y ωrY for some r > 0.

We need a preliminary result about the numerical Kodaira dimension whichwe introduced in Chapter 2.

Lemma 7.42. Let L be a nef line bundle on a smooth projective surface X wtihν(X,L) = 1. Then κ(X,L∗) = −∞.

Proof. Assume that H0(X,Lk) 6= 0 for some k < 0. Then either Lk is trivialor kc1(L) is represented by an effective divisor. The first case contradicts thatν(X,L) = 1, while the second contradicts that L is nef. The statement follows.

Theorem 7.43. [175, Thm. 1.6] Let X be a nonminimal smooth projective surface.If X is not a relatively minimal elliptic surface (Definition 6.9), any Fourier-Mukaipartner Y of X is isomorphic to X.

7.5. Derived categories and birational geometry 257

Proof. We use similar arguments as in the proof of Theorem 2.49. Let C be a (−1)-curve in X, that is, a smooth rational curve with C2 = −1. Then ω∗X |C ' OC(1) is

ample. Let us write T = p−1X (C) → Z. Since no curve can be contracted by the two

projections pX and pY , if D → T is contracted by pY , the induced map D → C

is finite. It follows that p∗Xω∗X |D is ample and the equality p∗Xω

rX ' p∗Y ωrY implies

that p∗Y ω∗Y |D is ample as well, which is impossible. Thus, pY |T : T → Y is a finite

morphism, so that dimT is either 1 or 2. If dimT = 1, one has dimZ = dim Z = 2and X ' Y by Proposition 7.19. If dimT = 2, one has projections pX |T : T → C

and pY |T : T → Y . Since ω−rX |C is ample, ω∗Y is nef by Lemma 2.43. Moreover,Lemma 2.44 yields ν(Y, ωY ) = ν(T, pY ∗|TωY ) = ν(T, pX∗|TωX) = ν(C,ωX |C) = 1,because ωX |C is anti-ample. Then ω∗X is nef and ν(X) = 1 by Theorem 2.49. ByLemma 7.42, κ(X) = −∞. Moreover, one has K2

X = 0 since ν(X) = 1. Thus, theminimal model X0 of X verifies κ(X0) = −∞ and K2

X0≤ 0, with equality only if

X is minimal. The classification of surfaces implies thus that X is either a minimalelliptic ruled surface or a rational ruled surface with invariant e = 2. Since X isnot minimal, only the second possibility may occur. Then X ' Y by Proposition7.22.

We know that there are only a finite number of nonisomorphic relative com-patified Jacobians JX/B(r, d) (Proposition 6.32). Since we also know that thereare a finite number of Fourier-Mukai partners of a K3 surface (Theorem 7.32) orof an Abelian surface (Corollary 7.39), this completes the proof of Theorem 7.17.

7.5 Derived categories and birational geometry

The condition that the derived categories of two surfaces are equivalent allowsone to characterize very precisely the relationship between the surfaces, basicallybecause minimal models of algebraic surfaces are completely classified. In higherdimensions, the situation is more involved. The so-called “minimal model pro-gram” has been completed in dimension three thanks to the work of Mori [221],who built on previous results by Reid, Kawamata, Kollar, Shokurov and others.A fundamental ingredient in Mori’s work is the reduction of the existence of flips,a rather difficult result, to the easier question of the existence of flops. These mayregarded as birational analogues of surgery in algebraic topology (one should notethat topological invariants of smooth projective varieties, such as Hodge numbers,are invariant under flops).

The crux of Bridgeland’s contribution to the topic is a moduli space inter-pretation of flops of smooth threefolds. This is accomplished by introducing finemoduli spaces of perverse sheaves; the integral functor associated with the relevantuniversal object is a Fourier-Mukai transform, i.e., it is an equivalence of derived


categories. This is the key to proving the following important theorems, due toBridgeland.

Theorem 7.44. [62, Thm. 1.1] Let X be a (complex) projective threefold with termi-nal singularities and f1 : Y1 → X, f2 : Y2 → X crepant resolutions of singularities.Then there is an equivalence of triangulated categories Db(Y1) ' Db(Y2).

Theorem 7.45. [62] Let X and Y be two birational smooth Calabi-Yau threefolds.Then X and Y are Fourier-Mukai partners, that is, there is an equivalence oftriangulated categories Db(X) ' Db(Y ).

In the course of this section we shall build proofs of these results.

7.5.1 A removable singularity theorem

We now generalize Bridgeland’s criterion for an integral functor to be an equiva-lence of categories. We assume that the base field k has characteristic zero.

Let us consider two projective varieties X and Y . We know from Proposition1.11 that a flat family of sheaves on Y parameterized by X may be characterizedas an object K• of the derived category Db(X × Y ) such that the object Lj∗xK•is a sheaf on Y for every (closed) point x ∈ X (where jx : x × Y → X × Y isas usual the natural immersion). More generally, any object K• of Db(X × Y ) offinite Tor-dimension over X can be thought of as a parameterization of objectsLj∗xK• of the derived category Db(Y ); finite Tor-dimension over X is needed toguarantee that the objects Lj∗xK• have bounded cohomology.

Moduli problems can be also formulated in terms of derived categories. LetD be a full subcategory of Db(Y ) changes with the property that any objectof Db(Y ) isomorphic to an object of D is also an object of D. Then a family ofobjects of D parameterized by an algebraic variety S is an object E• of Db(S×Y ) offinite Tor-dimension over S such that for every (closed) point s ∈ S, the complexLj∗sE• is an object of D. Two such families E•, F• are considered equivalent ifE• ' F• ⊗ π∗S(L) for a line bundle L on S. We can define a functor FD on thecategory of schemes by associating to an algebraic variety S the set of equivalenceclasses of families of objects of D parameterized by S. An algebraic variety M(FD)is a fine moduli space for D if there exists a family (a relative universal family) P•of objects of D parameterized by M(FD) such that for any algebraic variety S andany family E• of objects of D parameterized by S, there exists a unique morphismφ : S →M(FD) such that E• ' L(φ× Id)∗(P•)⊗ π∗S(L) for a line bundle L on S.In other words, the variety M(FD) represents the functor FD. Thus, there is anequivalence

Hom(S,M(FD)) ' FD(S)

φ 7→ L(φ× Id)∗(P•) .(7.8)


We will see in Section 7.5.2 an example of a functor defined by triangulatedsubcategories of the bounded derived categories — the functor of relative perversepoint sheaves — which is representable.

There is a Kodaira-Spencer map for families E• of objects of D parametrizedby an algebraic variety S. The definition is based on the tangent space to thefunctor, which we are going to describe.

Let D = Spec k[ε]/ε2 be the double point scheme and j0 : s0 → D theimmersion of its unique closed point. We recall that the tangent space TsS toan algebraic variety S at a point s can be defined as the space Hom(D,S)s ofthe scheme morphisms v : D → S mapping s0 to s. In the same vein, given anobject G• of D, the tangent space to the functor FD at the “point” [G•] is thespace T[G•]FD of classes of families F• ∈ FD(D) endowed with an isomorphism$ : Lj∗0F• ∼→ G• in the derived category (cf. [197, Def. 3.2.1]). Moreover, suchfamilies (F•, ϕ) are in a one-to-one correspondence with the extensions of G• byitself in the sense of the triangulated categories (cf. Definition A.69). This canbe easily seen by representing F• as a bounded complex of sheaves L• on D × Ywhich are flat over D; then $ : Lj∗0F• ∼→ G• induces an isomorphism j∗0L• ∼→ G•Moreover, if p denotes the projection D×Y → Y , the exact sequence of complexes

0→ j∗0L• → p∗L• → j∗0L• → 0 ,

and the morphism j∗0L• ∼→ G• define an extension of G• by itself as an object ofthe triangulated category Db(Y ).

Taking into account the identification between extensions and Hom1 groupsgiven by Proposition A.70, we have a one-to-one correspondence

T[G•]FD ' Hom1Db(Y )(G

•,G•) . (7.9)

Let now E• be a family of objects of D parametrized by an algebraic variety S.Any tangent vector v ∈ TsS ' Hom(D,S)s defines a morphism (v× Id) : D×Y →S × Y , whose composition with the immersion Y → D × Y , y 7→ (s0, y), is theinclusion js : Y ' s × Y → S × Y . Thus, L(v × Id)∗E• is a family of objectsof D parameterized by the double point scheme D equipped with an isomorphismLj∗0 (L(v×Id)∗E•) ∼→ Lj∗sE•. Therefore it defines as a point in TLj∗sE•FD. Accordingto Equation (7.9), we identify the latter space with Hom1

Db(Y )(Lj∗sE•,Lj∗sE•). In

this way we have a map

KSs(E•) : TsS → Hom1Db(Y )(Lj

∗sE•,Lj∗sE•)

v 7→ L(v × Id)∗E• ,(7.10)

called the Kodaira-Spencer map for the family E•.


Lemma 7.46. If there exists a fine moduli space M = M(FD) for D, and P•M isa relative universal family, the Kodaira-Spencer map

KSx(P•M ) : TxM → Hom1D(Y )(Lj

∗xP•M ,Lj∗xP•M ) ,

is an isomorphism for every (closed) point x ∈M .

Proof. Since M is a fine moduli space, we have an equivalence

TxM ' Hom(D,M)x ∼→ TLj∗xP•MFD ' Hom1D(Y )(Lj

∗xP•M ,Lj∗xP•M )

v 7→ L(v × Id)∗(P•M ) ,

which is precisely the Kodaira-Spencer map for the universal family P•M .

Proceeding as in the proof of Lemma 1.24, one has the following result.

Lemma 7.47. Let X be a projective variety and E• an object of Db(X × Y ) whichis a family of objects of D parameterized by X. The morphism

Hom1D(X)(Ox,Ox)→ Hom1

D(Y )(Lj∗xE•,Lj∗xE•)

induced by the integral functor ΦE•

X→Y coincides with the Kodaira-Spencer morphismfor the family E•.

In the rest of this section we make the following assumptions:

1. Y is a smooth projective algebraic variety of dimension m.

2. D is a full subcategory of Db(Y ) whose objects fulfil the following properties:

• They are simple, that is, one has HomD(Y )(E•, E•) = k for any objectE• of D.

• They are special (Definition 2.54), that is, E• ' E•⊗ωY for every objectE• of D.

• HomiD(Y )(E•, E•) = 0 for i < 0 and for every object E• of D.

• HomD(Y )(E•,F•) = 0 if E• and F• are non-isomorphic objects of D.

3. There is a projective algebraic variety M which is a fine moduli space M =M(FD) for D and an irreducible component j : X →M of dimension m.

Since Y is smooth and the objects E• are special, the third property listed inCondition 2 is equivalent by Serre duality to Homi

D(Y )(E•, E•) = 0 for i /∈ [0,m].

We give now a criterion for deciding when such a moduli space is smooth.This is based on a corollary of an “intersection theorem” in commutative algebra(cf. Corollary A.99). Let P•M be the universal family. We denote by P• = L(j ×IdY )∗(P•M ) its restriction to X × Y .


Theorem 7.48. If there is a closed subscheme Z → X×X of dimension d ≤ m+1such that for every (closed) point (x1, x2) /∈ Z one has

HomiD(Y )(Lj

∗x1P•, Lj∗x2

P•) = 0 for every i ∈ Z ,

then X is smooth and the integral functor ΦP•

X→Y : Db(X) → Db(Y ) is an equiva-lence of categories.

Proof. By Proposition 1.13, the integral functor Ψ = ΦP•∨⊗π∗Y ωY [m]

Y→X is a left ad-joint to Φ = ΦP

•

X→Y : Db(X) → Db(Y ). The composition Ψ Φ is then an integralfunctor whose kernel is the convolution M• = (P•∨ ⊗ π∗Y ωY [m]) ∗ P• of the twokernels.

Given two points x1, x2 of X, the adjunction between Φ and Ψ gives rise toisomorphisms

HomiD(X×X)(M•,O(x1,x2)) ' Homi

D(X)(Lj∗x1M•,Ox2)

' HomiD(Y )(Φ(Ox1),Φ(Ox2))

' HomiD(Y )(Lj

∗x1P•,Lj∗x2

P•)

for every integer i. By Proposition A.91, the support of the restriction of M• tothe complement U = (X ×X) −∆ of the diagonal is contained in (X ×X) − Zand so that it has codimension greater than or equal to m− 1. We can now applyCorollary A.97 to obtain that

hd(M•|U ) ≥ m− 1 . (7.11)

Since Y is smooth and the objects Lj∗xP• are special, by Serre duality, one hasthat

HommD(Y )(Lj

∗x1P•,Lj∗x2

P•) ' HomD(Y )(Lj∗x2P•,Lj∗x1

P•)∗ .By the last property in our second assumption, the latter group vanishes for x1 6=x2, and then hd(M•

|U ) ≤ m− 2; thus Equation (7.11) implies that M•|U = 0, so

that M• is (topologically) supported on the diagonal. Then one has

HomiD(Y )(Lj

∗x2P•,Lj∗x1

P•) = 0

unless x1 = x2 and i ∈ [0,m].

We now apply Corollary A.99 to prove that X is smooth. To do so, weconsider for every (closed) point x ∈ X the complex E•(x) = (Ψ Φ)(Ox) 'ΦM

•

X→X(Ox). Since HomiD(X)(E•(x1),Ox2) = 0 unless x1 = x2 and 0 ≤ i ≤ m, we

have to prove that H0(E•(x)) ' Ox. This will prove that X is smooth and thatE•(x) = (Ψ Φ)(Ox) ' Ox. The proof follows the same idea of that of Theorem1.27 with some modifications (cf. [71, Thm. 6.1]): we consider an exact triangle

C• → (Ψ Φ)(Ox) αx−−→ Ox → C•[1] , (7.12)


where αx is the adjunction morphism and C•[−1] is a cone of αx. By takinghomomorphisms in Ox and using that Ψ is a left adjoint to Φ, we get an exactsequence

0→ Hom0D(X)(Ox,Ox)→ Hom0

D(X)(Φ(Ox),Φ(Ox))→ Hom0D(X)(C•,Ox)

→ Hom1D(X)(Ox,Ox) τ−→ Hom1

D(X)(Φ(Ox),Φ(Ox))→ . . . (7.13)

and isomorphisms HomiD(X)(Φ(Ox),Φ(Ox)) ' Homi

D(X)(C•,Ox), for i < 0, so thatHomi

D(X)(C•,Ox) = 0 for i < 0. Moreover, by Lemma 7.47, the morphism τ isthe Kodaira-Spencer map for the universal family P•. Furthermore, the Kodaira-Spencer map for P• is the composition of the tangent map TxX → TxM withthe Kodaira-Spencer map for the universal family P•M . Since the tangent mapis injective because k has characteristic zero and the Kodaira-Spencer map forthe universal family P•M is an isomorphism by Lemma 7.46, we have that τ isinjective. It follows that Homi

D(X)(C•,Ox) = 0 for i < 0 so that Hi(C•) = 0 fori < 0 by Remark A.92. Taking cohomology in the exact triangle (7.12) one obtainsthat H0((Ψ Φ)(Ox)) ' Ox.

Now we know that X is smooth and that (ΨΦ)(Ox) ' Ox for every point x.We can apply Theorem 2.6 to prove that ΨΦ is fully faithful, since the skyscrapersheaves Ox form a spanning class for Db(X) (Proposition 2.52). By Remark 1.21,this implies that Φ is fully faithful as well so that Φ is an equivalence of categoriesby Corollary 2.56.

Flops

Flops are very simple instances of birational transformations. The precise defini-tions is the following:

Definition 7.49. A flop is a diagram

X

f @@@@@@@@ X+

f+||||||||

Y

where Y is a projective Gorenstein variety and f and f+ are crepant resolutionsof singularities (Definition 7.3) whose exceptional loci have codimension equal orgreater than 2. We also require the existence of a divisor D in X such that −Dis relatively f -ample in X and the strict transform D+ of D in X+ is relativelyf+-ample. 4

Since D+ is f+-ample, we have that f+ is isomorphic to the projective mor-phism X+ ' Proj

⊕s≥0 f

+∗ (OX+(sD+)) → Y which is actually isomorphic to


Proj⊕

s≥0 f∗(OX(sD)) → Y . This proves that f+ is completely determined by fand D.

By Lemma 7.5, if Y has terminal singularities, the condition on the excep-tional locus is automatically fulfilled.

The importance of flops is evident from the following result.

Proposition 7.50. [175, Lemma 4.6] Let α : X 99K Y be a crepant birational mapbetween projective threefolds with only terminal singularities. Then α may be de-composed into a sequence of flops.

t-structures

We now give some notions about triangulated categories that generalize what wesaw in Section 2.1.

Definition 7.51. Let A be a triangulated category. A full subcategory B ⊆ A isright admissible if the inclusion functor B → A has a right adjoint. 4

If B ⊆ A is a full subcategory, one can define the right orthogonal subcategoryas the full subcategory B⊥ of A whose objects are the objects a in A such that

HomiA(b, a) = 0 , for all objects b in B and all i ∈ Z .

One easily sees that if B ⊆ A is a right admissible triangulated subcategory, thenfor every object a in A there is an exact triangle

b→ a→ c→ b[1]

where b is an object in B and c is an object in B⊥. We then say that A admits asemi-orthogonal decomposition in terms of B and B⊥ and write A ' (B⊥,B).

The first example of a right admissible triangulated subcategory is given inthe next proposition. This example will be of great importance in this section.

Proposition 7.52. Let f : X → Y be a projective morphism of varieties such thatRf∗OX ' OY . Then the inverse image

Lf∗ : D(Y )→ D(X)

is fully faithful and makes D(Y ) into a right admissible triangulated subcategoryof D(X).

Proof. The functor Lf∗ has Rf∗ as a right adjoint. By the projection formula,the composition Rf∗ Lf∗ is the twist by Rf∗OX so that it is the identity by thecondition Rf∗OX ' OY . Then Lf∗ is fully faithful (see Section 1.3) and we canidentify D(Y ) with this image by Lf∗, and this is a right admissible triangulatedcategory.


If either Y is smooth or f is of finite Tor-dimension, then Lf∗ maps Db(X)to Db(Y ) and we can identify Db(X) with a right admissible triangulated categoryof Db(Y ).

Let A be a triangulated category.

Definition 7.53. A t-structure on A is a right admissible subcategory A≤0 of A

which is preserved by the shift functor, that is A≤0[1] ⊂ A≤0. 4

We will use the notations A≤i = A≤0[−i], A≥i = (A≤i−1)⊥, A<i = A≤i−1

and A>i = A≥i+1.

Definition 7.54. The heart (or core) of a t-structure A≤0 ⊂ A is the full subcategoryH = A≤0 ∩ A≥0. 4

One can prove [35] that the heart H of a t-structure is an Abelian category.An exact sequence

0→ a→ b→ c→ 0

in H is by definition an exact triangle a → b → c → a[1] in A whose vertices areobjects of H.

The first example of a t-structure is the standard t-structure on the derivedcategory D(A) of an Abelian category A. This is defined by taking D(A)≤0 as thefull subcategory defined by all complexes E• with no strictly positive cohomologyobjects, namely Hi(E•) = 0 for all i > 0. The heart of the standard t-structure isthe subcategory H of complexes E• such that Hi(E•) = 0 for all i 6= 0. There is anatural equivalence of Abelian categories A ' H.

The same applies to the bounded derived category Db(A).

Assume that B is another Abelian category and that there is an equivalenceof triangulated categories Φ: D(A) ∼→ D(B). The image by Φ of the standardt-structure on D(A) is a t-structure on D(B) and we can identify A with thefull subcategory of D(B) defined as the heart of this t-structure. This suggeststhat the study of the t-structures on a derived category D(B) is the right toolto determine all the Abelian categories A whose derived category is equivalent toD(B). This applies in particular for the derived category D(X) (or Db(X)) ofan algebraic variety: any Fourier-Mukai partner Y defines a t-structure in D(X)and the study of such t-structures is the natural tool for finding Fourier-Mukaipartners.

7.5.2 Perverse sheaves

In this section, f : X → Y will be a birational morphism of projective varietiessuch that Rf∗OX ' OY , and f has relative dimension 1, that is, no subvarieties


of dimension greater than one are contracted by f . The exceptional locus of f willbe denoted by E. It is a subscheme of X of dimension not exceeding one. We shallwrite V = Y − f(E) and U = f−1(U) = X −Y so that f induces an isomorphismf|U : U ∼→ V .

By Proposition 7.52, the inverse image Lf∗ makes D(Y ) into a right admissi-ble triangulated subcategory of D(X). Then we have a semi-orthogonal decompo-sition D(X) ' (C, D(Y )) where C = D(Y )⊥. The objects of C are the complexesE• in D(X) such that Rf∗E• = 0. They are supported on the exceptional locus off .

Lemma 7.55. An object E• of D(X) is in C if and only if all its cohomology sheavesHi(E•) are objects of C.

Proof. There is a spectral sequence Ep,q2 = Rpf∗(Hq(E•)) converging to Ep+q∞ =Hp+qRf∗(E•). Then Rf∗(E•) = 0 if all the sheaves Hq(E•) belong to C. For theconverse, since f has relative dimension 1, one has Ep,q2 = 0 for p 6= 0, 1. If thereis a nonzero element in Ep,q2 , it defines a cycle that survives to infinity and givesa nonzero element of Ep+q∞ = 0. It follows that Ep,q2 = 0 for all p and q.

Let D(X)≤0 be the standard t-structure on the derived category D(X). Itinduces a t-structure C≤0 = D(X)≤0 ∩ C on C. We also denote by D(Y )≤0 thestandard t-structure on D(Y ). Following [35] we can define for every integer p at-structure pD(X)≤0 in D(X) by letting

E• ∈ Ob(pD(X)≤0) if Rf∗(E•) ∈ Ob(D(Y )≤0) and

HomD(X)(E•,F•) = 0 for all F• ∈ Ob(C>p) ;

as a consequence, one has

E• ∈ Ob(pD(X)≥0) if Rf∗(E•) ∈ Ob(D(Y )≥0) and

HomD(X)(F•, E•) = 0 for all F• ∈ Ob(C<p) .

Definition 7.56. The category of p perverse sheaves for f is the heart

p Per(X/Y ) = pD(X)≤0 ∩ pD(X)≥0

of the above t-structure. When p = −1 we simply write

Per(X/Y ) = −1 Per(X/Y )

and call it the category of perverse sheaves. 4


As it often happens in mathematics, this terminology is actually a misnomer:perverse sheaves may fail to be sheaves. By its very definition, a complex E• inD(X) is a perverse sheaf if and only if it verifies three conditions:

1. Rf∗(E•) is a sheaf, i.e., it is a complex concentrated in degree zero.

2. HomD(X)(E•,F•) = 0 for all complexes F• with Rf∗F• = 0 and Hi(F•) = 0for i < 0.

3. HomD(X)(F•, E•) = 0 for all complexes F• with Rf∗F• = 0 and Hi(F•) = 0for i ≥ −1.

One should be aware that these are completely different objects from the “usual”constructible perverse sheaves (for these, see, e.g., [173, 35]). Especially in theliterature about stability conditions for derived categories (see Appendix D), itis somewhat standard to use “perverse (coherent) sheaf” to refer to an objectin the heart of a fixed nonstandard t-structure, regardless of the origin of thatt-structure.

The following result describes explicitly what perverse sheaves look like.

Lemma 7.57. A complex E• in D(X) is a perverse sheaf if and only if it satisfiesthe following conditions:

1. Hi(E•) = 0 unless i = −1 or i = 0.

2. R1f∗(H0(E•)) = 0 and f∗(H−1(E•)) = 0.

3. HomX(H0(E•),F) = 0 for any sheaf F in C.

Proof. If E• is a perverse sheaf, Rf∗(E•) is a sheaf. From the spectral sequenceEp,q2 = Rpf∗(Hq(E•)) =⇒ Ep+q∞ = Hp+qRf∗(E•), and the fact that Ep,q2 = 0 forp 6= 0, 1 (see the proof of Lemma 7.55), we deduce 2 and that Rf∗(Hi(E•)) = 0unless i = −1 or i = 0. Then, by Lemma 7.55, the truncated complex (E•)>0 isan object of C>0 and the truncated complex (E•)<−1 is an object of C<−1. Sincethere exist morphisms (E•)<−1 → E• and E• → (E•)>0, one has that (E•)<−1 = 0and (E•)>0 = 0, which proves 1. Part 3 follows because any nonzero morphism ofsheaves H0(E•)→ F gives a nonzero morphism E• → F in D(X).

For the converse, assume that the three conditions of the statement are ful-filled. Conditions 1 and 2 and the above spectral sequence prove that Rf∗E• is asheaf. If F• is an object of C≥0 and E• → F• is a nonzero morphism, then all thecohomology morphisms Hi(E•)→ Hi(F•) for i 6= 0 are automatically zero due tocondition 1, and thenH0(E•)→ H0(F•) cannot be zero. This contradicts condition3 because H0(F•) is an object of C by Lemma 7.55; then HomD(X)(E•,F•) = 0.Finally, if F• is an object of C<−1, HomD(X)(F•, E•) = 0 by condition 1.


Corollary 7.58. Let E• be a perverse sheaf. If Rf∗(E•) = 0, then E•[−1] is a sheaf.

Proof. E• is an object of C, so that H0(E•) is also an object of C by Lemma7.55. Condition 3 of Lemma 7.57 implies then that H0(E•) = 0; thus E•[−1] is asheaf.

An elementary consequence of Lemma 7.57 is that the structure sheaf OX isa perverse sheaf. Now we consider exact sequences

0→ I• → OX → E• → 0 (7.14)

in the Abelian category Per(X/Y ) of perverse sheaves. We adopt the followingdefinition.

Definition 7.59. The perverse sheaf E• is called a perverse structure sheaf and theperverse sheaf I• is called the corresponding perverse ideal sheaf. 4

Lemma 7.60. Perverse ideal sheaves are actually sheaves, that is, Hi(I•) = 0 fori 6= 0.

Proof. The result is obtained by taking cohomology in Equation (7.14).

The Euler characteristic of two objects of Db(X) was defined (cf. Eq. (1.5))when X is a smooth projective variety. However, the formula

χ(L, E•) =∑i

(−1)i dim HomiD(X)(L, E•)

makes sense for any projective variety if L is a locally free sheaf and E• is an objectof Db(X).

Definition 7.61. Let X be a projective variety. Two objects E• and F• of Db(X)are derived numerically equivalent if for any locally free sheaf L on X the Eulercharacteristics χ(L, E•) and χ(L,F•) coincide. 4

When X is smooth, E• and F• are derived numerically equivalent if and onlyif they have the same Chern characters. For sheaves, derived numerical equivalencehas some elementary properties.

Lemma 7.62. Let X be a projective variety and E a coherent sheaf on X.

1. E is derived numerically equivalent to zero if and only if E = 0.

2. E is derived numerically equivalent to the structure sheaf of a closed point ifand only if E ' Ox for some closed point x ∈ X.


Proof. For n 0, HomiD(X)(OX(−n), E) = 0 for i 6= 0, so that χ(O(−n), E) =

dim Hom0D(X)(OX(−n), E); moreover, there is a surjective morphism

Hom0D(X)(OX(−n), E)⊗k OX → E(n)→ 0 .

Thus, if E is derived numerically equivalent to zero, one has E = 0. If E is de-rived numerically equivalent to the structure sheaf of a closed point, one hasχ(L, E) = rk(L) for any locally free sheaf L. Reasoning as above, there is a surjec-tive morphism OX → E(n) → 0, and then E(n) is the structure sheaf of a closedsubscheme Y of X. Moreover E(n) is also derived numerically equivalent to thestructure sheaf of a closed point. If x ∈ Y is a closed point, the kernel of thesurjection OY → Ox → 0 is a sheaf derived numerically equivalent to zero, andthen OY ' Ox.

We now give the following definition:

Definition 7.63. A perverse point sheaf is a perverse structure sheaf which is de-rived numerically equivalent to the structure sheaf Ox of a closed point. 4

Example 7.64. If x ∈ U is a point in the open subset where f is an isomorphism,then Ox is a perverse point sheaf. However if x ∈ E is a point of the exceptionallocus, this may fail to be true. Assume for instance that the exceptional locus off is a smooth rational curve i : E → X defined by an ideal sheaf JE . For everypoint x ∈ E there is an exact sequence

0→ JE → mx → i∗mE,x → 0 ,

where mE,x denotes the ideal of x in the curve E. Since mE,x ' OE(−1), wesee that Rf∗i∗mE,x = 0 and then i∗mE,x is an object of C. By Lemma 7.57, theexistence of the nonzero morphism mx → i∗mE,x implies that mx is not a perversesheaf. Thus, the skyscraper sheaf Ox is not a perverse point sheaf. 4

Lemma 7.65. If E• is a perverse point sheaf, then

Rf∗E• ' Oy

for a (closed) point y ∈ Y .

Proof. By adjunction, the sheaf Rf∗Ox is derived numerically equivalent to thestructure sheaf of a closed point, so that it is isomorphic to the structure sheaf ofsome closed point y ∈ Y by Lemma 7.62.

Lemma 7.66. Let E• and F• be perverse point sheaves on X. Then

HomD(X)(E•,F•) '

k if E• ' F•

0 otherwise.


Moreover, one has HomiD(X)(E•, E•) = 0 for i < 0 and every perverse point sheaf

E•.

Proof. If E• is a perverse point sheaf, Rf∗(E•) ' Oy for a point y ∈ Y by Lemma7.65. Then HomD(X)(OX , E•) ' HomD(Y )(OY ,Oy) ' k and Homi

D(X)(OX , E•) 'Homi

D(Y )(OY ,Oy) = 0 for i < 0. Taking homomorphisms of the exact sequence(7.14) of perverse sheaves into F• gives

0→ HomD(X)(E•,F•)→ HomD(X)(OX ,F•) ' k .

Then, if HomD(X)(E•,F•) 6= 0, the surjective morphism OX → F• of perversesheaves factors trough a surjective morphism φ : E• → F• of perverse sheaves.The kernel kerφ in the category Per(X/Y ) is derived numerically equivalent tozero, so that Rf∗ kerφ is a sheaf of Y numerically equivalent to zero. By Lemma7.62, Rf∗ kerφ = 0. By Corollary 7.58, kerφ[−1] is a sheaf. Since it is numericallyequivalent to zero, it has to be zero by Lemma 7.62, so that φ is an isomorphism.This proves the first part. For the second, taking derived homomorphisms of I•into the exact sequence (7.14) of perverse sheaves, and using the fact that I• isa sheaf (Lemma 7.60), we obtain Homi(I•, E•) = 0 for i < 1. Moreover, sinceHomi

D(X)(OX , E•) = 0 for i < 0, by taking homomorphisms of the exact sequence(7.14) into E• one has Homi

D(X)(E•, E•) = 0 for i < 0 as claimed.

Our next aim is to construct an algebraic variety W and a projective mor-phism f+ : W → Y (which under suitable assumptions will be a flop of f) suchthat the closed points of W parameterize perverse point sheaves on X. We willnot enter into the details of the proof of the existence of the moduli space W

(see [62]); however we need to state some of the natural base change propertiesof perverse sheaves which makes the construction of this moduli space possible,because they are important on their own and will be useful for other purposes.Let S be a scheme, which will play the role of the base scheme of the space ofparameters for a family. We have for every point s ∈ S a commutative diagram

X js //

f

S ×X

fS

Y

js // S × Y

where fS = 1 × f . Let us denote by πS the projection S ×X → S as in Section1.2.1.

Definition 7.67. A family of perverse sheaves (or a relative perverse sheaf) forf : X → Y over S is an object E• in D(S × X), flat over S, such that for everypoint s, the complex Lj∗sE• of D(X) is a perverse sheaf for f . Two such familiesE•, F• are equivalent if F• ' E• ⊗ π∗SL for some line bundle L on S. 4


If φ : T → S is a morphism of schemes, and E• is a relative perverse sheaffor f over S, the derived pullback L(φ× 1)∗(E•) is a relative perverse sheaf for fover T .

Proposition 7.68. Let E• ∈ Ob(D(S × X)) be a family of perverse sheaves for fover S. Then the complex RfS∗E• is a sheaf G on S×Y flat over S and one has anisomorphism of sheaves Gs ' Rf∗(Lj∗sE•) for every point s ∈ S, where Gs = j∗sG.

Proof. One has Lf∗S(js∗(OY )) ' js∗(OX). Then, the projection formula gives

js∗(Rf∗(Lj∗sE•)) ' RfS∗(js∗(Lj∗sE•)) ' RfS∗(js∗(OX)L⊗E•))

' js∗(OY )L⊗RfS∗(E•) ' js∗(Lj∗s (RfS∗(E•))) .

Since Lj∗sE• is a perverse sheaf on X for any s ∈ S, Rf∗(Lj∗sE•) is a sheaf onY for any s ∈ S by Lemma 7.57. It follows that Lj∗s (RfS∗(E•)) is also a sheaffor any s ∈ S, and then RfS∗(E•) is a sheaf G on S × Y flat over S such thatGs ' Rf∗(Lj∗sE•) by Proposition 1.11.

We define the functor of relative perverse point sheaves as the functor whichassigns to a scheme S the set of exact triangles

I• → OS×X → E• → I•[1] ,

where for every point s ∈ S, the restriction Lj∗sE• is a perverse point sheaf. Byabuse of language we refer to E• as a relative perverse point sheaf, or a familyof perverse point sheaves. Notice that the restriction Lj∗sI• is a perverse idealsheaf; since perverse ideal sheaves are sheaves by Lemma 7.60, by Proposition1.11, I• ' I is a sheaf flat over S. The following result then follows.

Lemma 7.69. A relative perverse point sheaf E• is of finite Tor-dimension over S.

There is an existence theorem for the moduli space.

Theorem 7.70. [62, Theorem 3.8] The functor which assigns to a scheme S theset of equivalence classes of families of perverse point sheaves for f over S isrepresentable by a projective scheme M(X/Y ).

So M(X/Y ) is a fine moduli space of perverse point sheaves and there isa universal perverse point sheaf P• in D(M(X/Y ) × X) for f over M(X/Y ),of finite Tor-dimension over M(X/Y ), such that the perverse point sheaf on X

corresponding to a point w ∈ M(X/Y ) is the object Lj∗wP•, where jw : X 'w × X → M(X/Y ) × X is the natural embedding. More generally, given arelative perverse point sheaf E• ∈ Ob(D(S × X)) for f over S, there exists a


unique scheme morphism φ : S →M(X/Y ) such that E• ' L(φ× 1)∗P• ⊗ π∗S(L)for a line bundle L on S.

By Proposition 7.68, if E• ∈ Ob(D(S × X)) is a family of perverse pointsheaves for f over S, then RfS∗E• is a sheaf G on S×Y flat over S and one has anisomorphism of sheaves Gs ' Rf∗(Lj∗sE•) for every point s ∈ S, where Gs = j∗sG.Since Rf∗(Lj∗sE•) ' Oy for a point y ∈ Y , we see that, up to twisting by a linebundle coming from S, the sheaf G is the structure sheaf of the graph of a uniquemorphism of schemes g : S → Y , that is, one has

RfS∗E• ' Γg∗OS ⊗ p∗SL

for some line bundle L on S, where Γg : S → S × Y is the graph of g and pS : S ×Y → S is the projection. In particular, the universal perverse point sheaf P• givesrise to a morphism

f+ : M(X/Y )→ Y ,

uniquely characterized by the condition

RfM(X/Y )∗P• ' Γf+∗OM(X/Y ) ⊗ p∗M(X/Y )L (7.15)

for some line bundle L on M(X/Y ). The effect of f+ on closed points is simplythat of taking the derived direct image, that is, if E• is a perverse point sheaf onX and [E•] ∈M(X/Y ) is the closed point determined by it, then

f+([E•]) = y , with Oy ' Rf∗E•.

Now, twisting P• by π∗M(X/Y )(L)−1 we can normalize the universal relative per-verse point sheaf P• so that

RfM(X/Y )∗P• ' Γf+∗OM(X/Y ) . (7.16)

The fact that f is birational implies that f+ is birational as well. Actually, sincef induces an isomorphism U ∼→ V = f(U) on the complement U of the exceptionallocus, the skyscraper sheaf Ox is a perverse point sheaf (Example 7.64) and it isthe unique perverse point sheaf E• such that f+([E•]) = f(x), so that f+ gives anisomorphism

f+|(f+)−1(V ) : (f+)−1(V ) ∼→ V .

We can make this construction more algebraic by noting that the inverse of f+ overV is given by the morphism φ : V →M(X/Y ), y 7→ Of−1(y), corresponding by theuniversal property ofM(X/Y ) (Theorem 7.70) to the structure sheaf Γg∗(OV ) ofthe graph of the morphism g : V → X given as the composition of f−1 : V ∼→ U

and the immersion i : U → X.

Let W be the irreducible component ofM(X/Y ) containing (f+)−1(V ). Onecan prove that actually W = M(X/Y ) [62], though we do not need this result.


If we still denote by f+ the restriction of f+ : M(X/Y ) → Y to W , we have acommutative diagram of birational morphisms

X

f @@@@@@@ W

f+~~

Y

(7.17)

Remark 7.71. Let us denote again by P• the restriction to W ×X of the universalperverse point sheaf. There is a universal exact triangle

I → OW×X → P• → I[1]

which defines the universal perverse ideal sheaf I. As we have already observed,I is flat over W . Moreover, Chen has proved in [87, Prop. 4.2] that I → OW×X isinjective, and that I is actually the ideal of the fiber product X ×Y X →W ×X.Thus, P• is isomorphic to the structure sheaf OW×YX and there is a universalexact sequence of coherent sheaves

0→ I → OW×X → P• ' OW×YX → 0 .

Though I and OW×X are flat over W , the sheaf OW×YX is not, and this ex-plains why for some points w ∈W , the corresponding perverse point sheaf P•w 'Lj∗wOW×YX is a complex and not a sheaf. 4

7.5.3 Flops and derived equivalences

We now apply the results and notation of Section 7.5.2 to a crepant resolution ofsingularities f : X → Y of a projective threefold with Gorenstein terminal singu-larities (Definition 7.2). The morphism f satisfies Rf∗(OX) ' OY and contractsonly a finite number of curves. If Y ′ is the image of the exceptional locus, theanti-image Z of Y ′ by the fiber product morphism W ×Y W → Y is a surface.

Let us denote by P• the restriction to D(W × X) of the relative universalperverse sheaf. Since it is of finite Tor-dimension over W by Lemma 7.69, we canconsider the associated integral functor

Φ = ΦP•

W→Y : Db(W )→ Db(X) ,

described as Φ(F•) = RπX∗(π∗W (F•)L⊗P•).

Theorem 7.72. W is smooth, f+ is crepant and Φ is a Fourier-Mukai functor.

Proof. We use the removable singularity Theorem 7.48. By Lemma 7.65, the objectP•w = Lj∗wP• is simple and then its support is connected. Since Rf∗(P•w) ' Oy


with y = f+(w), the support of P•w is contained in the fiber f−1(y). Then P•w 'P•w ⊗ωX because f is crepant. Moreover HomD(X)(P•w1 ,P•w2) = 0 for w1 6= w2

and HomiD(X)(P•w,P•w) = 0 for i < 0 and every point w, by Lemma 7.66. Now,

if w1 and w2 are distinct points of W , one sees that

HomiD(X)(P•w1 ,P•w2) = 0

for all i if f+(w1) 6= f+(w2), that is, if (w1, w2) /∈ Z. Since dim Z = 2 ≤ 4,Theorem 7.48 implies that W is smooth and Φ is an equivalence.

Let us now see that f+ is crepant. For each point y ∈ Y , Φ induces an equiv-alence of categories between the full subcategory Df−1(y)(X) ⊂ Db(X) of objectstopologically supported on the fiber f−1(y) and the full sucategory D(f+)−1(y)(W )⊂ Db(W ) of complexes topologically supported on the fiber (f+)−1(y). Since ωX istrivial on an open neighborhood of f−1(y), the triangulated category Df−1(y)(X)has trivial Serre functor, and then D(f+)−1(y)(W ) has trivial Serre functor as well.By Proposition 7.8, f+ is crepant.

Consider the diagram

W ×XπW

vvvvvvvvv

πX //

fW

X

f

W W × Y

πWoo πY // Y

Then, by the projection formula and Equation (7.16), we have

Rf∗Φ(F•) ' RπY ∗RfW∗(π∗W (F•)L⊗P•) ' RπY ∗(π∗W (F•)

L⊗RfW∗P•))

' RπY ∗(π∗W (F•)L⊗Γf+∗OW )) ' Rf+

∗ F• ,

for any object F• of D(W ). Thus, there is a commutative diagram of exact functors

D(W ) Φ //

Rf+∗ $$HHHHHHHHH

D(X)

Rf∗vvvvvvvvv

D(Y )

(7.18)

We are now going to prove that the diagram (7.17) is a flop. We need apreliminary result.

Lemma 7.73. Let F• be an object of D(W ) such that Rf+∗ (F•) = 0. Then F• is

WIT−1 with respect to Φ if and only if it is a sheaf F• ' F on W .


Proof. Assume first that F• is WIT−1 so that Φ(F•) ' E [1] for a sheaf E on X.Then Rf∗(E [1]) = 0 by Equation (7.18). This implies that E is supported on theexceptional locus of f , so that E ⊗ ωX ' E because f is crepant. It follows that

HomiD(X)(E [1],OX) ' Hom3−i

D(X)(OX , E [1])∗ ' Hom3−iD(Y )(OY ,Rf∗(E [1]))∗ = 0

for every integer i. Thus, for any (closed) point w ∈ W , from the exact sequence0→ Iw → OX → P•w → 0 in the category of perverse sheaves, one obtains

HomiD(X)(E [1],P•w) ' Homi

X(E [1], Iw) = 0 unless 0 ≤ i ≤ 3 ,

together with 0→ HomD(X)(P•w, E [1])→ HomD(X)(OX , E [1]) = 0. Then one hasthe following vanishing result:

Hom3D(X)(E [1],P•w) ' Hom0

D(X)(P•w, E [1]) = 0 .

The Parseval formula HomiD(W )(F•,Ow) ' Homi

D(X)(E [1],P•w) (cf. Proposition1.34) implies that F• has homological dimension smaller than or equal to 2, andit is then isomorphic in D(W ) to a complex 0 → Lm−2 → Lm−1 → Lm → 0of coherent locally free sheaves (cf. Definition A.93). The cohomology sheavesHj(F•) are supported on curves, actually contracted by f ; if Z is any irreduciblecomponent of one of these curves and z0 is its generic point, the stalk OW,z0 isa local regular Noetherian ring of dimension 2. Applying the acyclicity LemmaA.95 to the complex Lm−2

z0 → Lm−1z0 → Lmz0 of free OW,z0 -modules, we have that

Hm−j(F•)z0 = 0 for j > 0. Then Hm−j(F•) = 0 for j > 0 and F• ' H0(F•) is asheaf.

We now prove the converse. Suppose that F• is a sheaf F on W . As aboveRf∗(Φ(F)) = 0 by Equation (7.18) and then Rf∗(Φi(F)) = 0 for every i byLemma 7.55. Let us denote by n0 and n1 the minimum and the maximum of theintegers n such that the n-th cohomology sheaf Hn = Φn−1(F) of Φ(F)[−1] is notzero. We then have nonzero groups

Hom−n1D(X)(Φ(F)[−1],Hn1) ' HomD(X)(Φ(F)[−1],Hn1 [−n1])

Homn0D(X)(H

n0 ,Φ(F)[−1]) ' HomD(X)(Hn0 [−n0],Φ(F)[−1]) .

If Φ(F)[−1] is not a sheaf, one has either n0 < 0 or 0 < n1, and we can finda positive integer n and a sheaf H verifying Rf∗H = 0 and such that eitherHom−nD(X)(H,Φ(F)[−1]) 6= 0 or Hom−nD(X)(Φ(F)[−1],H) 6= 0. By the first part,the object Φ−1(H[1]) is a sheaf G on W . Then either Hom−nD(W )(G,F) 6= 0 orHom−nD(X)(F ,G) 6= 0 which is absurd because F and G are sheaves.

Proposition 7.74. The diagram (7.17) is a flop, or in other words, f+ : W → Y isa flop of f : X → Y .

7.6. McKay correspondence 275

Proof. We have to prove that if D+ is a divisor in W such that −D+ is relativelyf+-nef, then the the strict transform D in X is relatively f -nef. Let Z be a rationalcurve contracted by f+. If E• = Φ(OW (D+)) and F• = Φ(OZ(−1)), we have

χ(E•,F•) = χ(OW (D+),OZ(−1)) = χ(OZ(−1−D+ · Z)) = −D+ · Z ≥ 0 ,

by the Parseval formula (Proposition 1.34). On the complementary U of the excep-tional locus, E• is isomorphic to OX(D) and then c1(E•) = [D]. By Lemma 7.73,F•[−1] is a sheaf G and since Rf∗G = 0, the support of G is curve Z ′ contractedby f . Thus the Chern characters of G are ch0(G) = 0, ch1(G) = 0, ch2(G) = Z ′

and ch3(G) = 0, where the last equality follows from Riemann-Roch for f aftertaking into account that Rf∗G = 0 and that f is crepant. Moreover Z ′ · c1(X) = 0because Z ′ is contracted by f and f is crepant, so that Riemann-Roch gives

χ(E•,F•) = −χ(E•,G) = D · Z ′

Then D · Z ′ ≥ 0 and D is f -nef.

Proof of Theorem 7.44. By Proposition 7.50, the crepant birational map

X1

f1 AAAAAAAA X2

f2~~

Y

is decomposed into a sequence of flops. By applying Proposition 7.74 and Theorem7.72 we see that the derived categories Db(Y1) and Db(Y2) are equivalent.

Proof of Theorem 7.45. We proceed as above, taking into account that if X andY are Calabi-Yau threefolds, any birational map between them is crepant.

7.6 McKay correspondence

The classical McKay correspondence relates representations of a finite subgroupG ⊂ SL(2,C) to the cohomology of the minimal resolution of the Kleinian singu-larity C2/G, cf. [127]. An important application of the theory of integral functorsis a deep generalization of this correspondence due to Bridgeland, King and Reid[68]. In this section we review this result. Part of the material regarding equivariantderived categories has been taken from [250].

The starting point for the derived McKay correspondence is a smooth pro-jective complex variety Y acted on by a finite group G of automorphisms in sucha way that the canonical line bundle ωY is locally trivial as a G-sheaf, that is,there is a covering of Y by G-invariant open neighborhoods each of which carry a


nonvanishing G-invariant section of ωY . Then the quotient variety Y/G has onlyGorenstein singularities and the McKay correspondence can be generalized as anequivalence between the G-equivariant derived category of Y , and the ordinary de-rived category of a crepant resolution of singularitiesW → Y/G,Db(W ) ' DG(Y ).This equivalence is a G-equivariant version of an integral functor in a sense we aregoing to describe.

We restrict ourselves to the projective case, while the reader is referred to[68] for the quasi-projective situation.

7.6.1 An equivariant removable singularity theorem

The removable singularity theorem given in Section 7.5.1 can be straightforwardlyextended to the linearized case. We outline here the main results, whose proofsare completely analogous to the corresponding ones in the ordinary case.

We now consider an algebraic variety Y acted on by a finite group G anda full subcategory D of DG,b(Y ) with the property that any object of DG,b(Y )isomorphic to an object of D is also an object of D. A family of objects of D

parameterized by S is an object E• of De×G,b(S × Y ) of finite Tor-dimensionover S such that for every (closed) point s ∈ S, the complex Lj∗sE• is an objectof D. As in the ordinary case, two such families E•, F• are considered equivalentif E• ' F• ⊗ π∗S(L) in De×G,b(S × Y ) for a line bundle L on S. We can definea functor FD on the category schemes by associating to an algebraic variety S

the set of equivalence classes of families of objects of D parameterized by S. Thetangent space to FD at an object [G•] of D is now given by

T[G•]FD ' Hom1DG,b(Y )(G

•,G•) .

We also have a linearized Kodaira-Spencer map

KSGs (E•) : TsS → Hom1DG,b(Y )(Lj

∗sE•,Lj∗sE•) ,

where s is a point in a scheme S parameterizing a family E• of objects of D.If there exists a fine moduli space M = M(FD) for D, and P•M is a relativeuniversal family, by proceeding as in the proof of Lemma 7.46, we see that thelinearized Kodaira-Spencer map KSGx (P•M ) is an isomorphism for every (closed)point x ∈M(FD). As in Section 7.5.1, we assume that this is indeed the case, andconsider an irreducible component j : X →M = M(FD) of the fine moduli spacefor D.

We denote by P• = L(j×IdY )∗(P•M ) the restriction to X×Y of the relativeuniversal family P•M . P• is an object of De×G,b(X × Y ) flat over X. One canconsider the associated equivariant integral functor

ΦP•,e×G

X→Y : Db(X)→ DG,b(Y ) .

7.6. McKay correspondence 277

Proceeding as in the proof of Proposition 7.48, one has the following equivariantremovable singularity result.

Theorem 7.75. Suppose that Y is a smooth projective algebraic variety of dimen-sion m = dimX and that the following properties are fulfilled:

1. One has

HomDG,b(Y )(Lj∗x1P•,Lj∗x2

P•) '

0 for x1 6= x2

k for x1 = x2

for every pair of (closed) points x1 and x2 in X.

2. Lj∗xP• ' Lj∗xP• ⊗ ωY in DG,b(Y ) for every (closed) point x ∈ X, where ωYis equipped with its natural linearization (cf. Example 1.35).

3. HomiDG,b(Y )(Lj

∗xP•,Lj∗xP•) = 0 for i < 0 and any (closed) point x ∈ X.

If there is a closed subscheme Z → X ×X of dimension d ≤ m+ 1 such that forevery (closed) point (x1, x2) /∈ Z one has

HomiDG,b(Y )(Lj

∗x1P•, Lj∗x2

P•) = 0 for every i ∈ Z ,

then X is smooth and the equivariant integral functor

ΦP•,e×G

X→Y : Db(X)→ DG,b(Y )

is an equivalence of categories.

7.6.2 The derived McKay correspondence

Let G be a finite group acting on a smooth complex projective variety Y . Weassume that the canonical line bundle ωY is locally trivial as an equivariant sheaf(cf. Example 1.35), that is, every point of Y has an open G-invariant neighborhoodU such that there is an equivariant isomorphism OU ' ωU . This implies that thequotient variety Y/G has only Gorenstein singularities.

We are going to construct a crepant resolution of Y/G whose bounded derivedcategory is equivalent to the G-linearized derived category DG,b(Y ) of Y , usingequivariant integral functors. A candidate for such a resolution is Nakamura’s G-Hilbert scheme (cf. [232]), which parameterizes G-clusters on Y ; by a G-cluster onY we mean a zero dimensional G-invariant subscheme Z → Y such that Γ(Z,OZ)is isomorphic, as a G-vector space, to the regular representation C[G] of G. Thelength of a G-cluster is then the order ](G) of the group G, and any free orbitof G is a G-cluster. The construction of this G-Hilbert scheme can be outlined asfollows.


We consider the functor associating to a scheme S the set of closed G 'e ×G-invariant subschemes Z → S × Y , flat over S, such that for every closedpoint s ∈ S, the fiber Zs → Y is a G-cluster. Here S is equipped with the trivialaction of e. This functor is representable by a closed subscheme HilbG(Y ) ofthe Hilbert scheme of zero dimensional subschemes of Y of length ](G). There isa universal relative G-invariant closed subscheme

ZY → HilbG(Y )× Y ,

which is flat over HilbG(Y ).

There is a Hilbert-Chow morphism τ : HilbG(Y )→ Y/G which sends a closedpoint z ∈ HilbG(Y ) to the orbit supporting the corresponding G-cluster Zz. It isa surjective projective morphism and it is birational on one component.

Let W be the irreducible component of HilbG(Y ) containing the points thatcorrespond to the free orbits of G. We consider this component because it is notknown in general whether HilbG(Y ) is irreducible. We denote by Z = Z|W×Y therestriction of the universal relative G- invariant closed subscheme, which is finiteand flat over W .

If we consider the trivial group e acting on W , the structure sheaf OZ ofZ defines an equivariant kernel in De×G,b(W × Y ), flat over W . We then havean equivariant integral functor

ΦOZ ,e×GW→Y : Db(W )→ DG,b(Y )

E• 7→ ΦOZ ,e×GW→Y (E•) = RπGY,∗(π∗WE• ⊗OZ) .

We denote again by τ : W → Y/G the restriction to W of the Hilbert-Chowmorphism, which is a birational morphism. Let W ×Y/G W be the fiber productwith respect to τ .

The derived McKay correspondence is the following result [68, Thm. 1.1].

Theorem 7.76. Suppose that dim(W×Y/GW ) ≤ 1+dimY . The equivariant integralfunctor ΦOZ ,e×GW→Y : Db(W )→ DG,b(Y ) is an equivalence of categories. Moreover,W is smooth and τ : W → Y/G is a crepant resolution of singularities of thequotient variety Y/G.

Proof. We prove that ΦOZ ,e×GW→Y is an equivalence of categories and that W issmooth by applying Theorem 7.75 for the triangulated subcategory D ⊂ DG,b(Y )of all G-clusters in Y . In our situation P• = OZ , which is flat over W , and thenLj∗wP• ' j∗wOZ ' OZw for every point w ∈W .

Since HomiDG,b(Y )(OZw1

,OZw2) = 0 for all integers i if (w1, w2) /∈W×Y/GW ,

the dimension condition of Theorem 7.75 is satisfied by hypothesis.


We then need only to check that G-clusters in Y fulfil all the requirementsof Theorem 7.75. Condition 3 is automatic. For Condition 1, one first notices thatsince Γ(Zw,OZw) ' C[G] as G-vector spaces, then HomG

Y (OZw ,OZw) ' C[G]G 'C. If w1 and w2 are distinct points of W , the G-clusters Zw1 and Zw2 are different,and then any equivariant morphism ϕ : Zw1 → Zw2 must vanish at the points ofZx1 which are not in Zw2 ; since the equivariant sections are constant, this forcesϕ = 0. For Condition 2, we have to see that OZw ⊗ ωY ' OZw as G-linearizedsheaves for every G-cluster OZw , and this follows because ωY is trivial as a G-linearized sheaf on an open neighborhood of every orbit of G.

To finish the proof, we have only to see that τ : W → Y/G is crepant. Theproof is similar to the proof that f+ is crepant in Theorem 7.72. For each pointx ∈ Y/G, the equivalence ΦOZ ,e×GW→Y gives an equivalence of categories betweenthe full subcategory Dτ−1(x)(W ) ⊂ Db(W ) of objects topologically supported onthe fiber τ−1(x) and the full subcategory DG

π−1(x)(Y ) ⊂ DG,b(Y ) of G-linearizedcomplexes topologically supported on the fiber π−1(x) of the quotient morphismπ : Y → Y/G. Since ωY is trivial as a G-linearized sheaf on an open neighborhoodof π−1(x), the triangulated category DG

π−1(x)(Y ) has trivial Serre functor, andthen Dτ−1(x)(W ) also has trivial Serre functor. Proposition 7.8 implies that τ iscrepant.

Remark 7.77. When dimY ≤ 3, the condition on the dimension of the fiber prod-uct in the statement of this theorem holds true because the dimension of theexceptional locus of W → Y/G is less than or equal to 2. 4


Fourier-Mukai partners of K3 surfaces. If X and Y are K3 surfaces which areFourier-Mukai partners, the respective Hilbert schemes of n points, X [n] and Y [n],are Fourier-Mukai partners as well [250]. It is interesting to note the existence ofHilbert schemes X [n] and Y [n], where X and Y are surfaces, which are Fourier-Mukai partners but are not birational. An example is given by Markman [210].

Theorem 7.26 says that a Fourier-Mukai partner Y of a K3 surface X is amoduli space of stable sheaves onX. Huybrechts [154] proved a stronger statement,namely, that Y is isomorphic to a moduli space of µ-stable sheaves.

Fourier-Mukai partners of Abelian surfaces. Let Kum(A) be the Kummer surfaceassociated to the Abelian surface A. One defines the set K(Kum(A)) of isomor-phism classes of Abelian surfaces B such that Kum(B) ' Kum(A). Hosono, Lian,Oguiso and Yau [149] proved that FM(A) = K(Kum(A)).

Fourier-Mukai partners of bielliptic surfaces. Bridgeland and Maciocia have provedin this case a stronger statement than Theorem 7.17, namely, that a bielliptic


surface has no other partner than itself [70].

Derived categories and birational geometry. Kawamata, using a different ap-proach, has proved Bridgeland’s Theorem 7.44 for orbifolds [175]. Other significantcontributions are due to Chen [87] and Van den Bergh [289]. For a survey, we referto Bridgeland’s ICM address [63] and to Rouquier [260].

Derived categories, integral functors and string theory. In recent years, integralfunctors have found several applications in string theory. The most notable exam-ple is Kontsevich’s homological mirror symmetry conjecture [188]. This predicts anequivalence between the bounded derived category of coherent sheaves on a Calabi-Yau manifold and the Fukaya category of the mirror dual manifold [115, 116, 117].The conjecture implies a correspondence between self-equivalences of the derivedcategory and certain symplectic self-equivalences of the mirror manifold. Evidencein this direction has been provided in [96, 6, 148], among others.

Appendix A

Derived and triangulated

categories

by Fernando Sancho

A.1 Basic notions

We assume that the reader is familiar with the basics of category theory, as ex-pounded for instance in Mac Lane’s standard textbook [201]. Nevertheless, mainlyin order to fix notation and terminology, we recall here a few notions.

A category C consists of the following set of data:

1. a class Ob(C), whose elements are called the objects of C;

2. for each ordered pair of objects A,B ∈ Ob(C), a class HomC(A,B), whoseelements are called morphisms from A to B and denoted f : A→ B;

3. for each ordered triple of objects A,B,C, a map

HomC(B,C)×HomC(A,B)→ HomC(A,C)

(f, g) 7→ f g ,

called the composition map.

One requires that the composition is associative and that for any object A thereexists the identity morphism IdA ∈ HomC(A,A), satisfying f = IdA f for anyf ∈ HomC(B,A) and g = g IdA for any g ∈ HomC(A,B).

282 Appendix A. Derived and triangulated categories

A category is said to be small if the classes of both its objects and its mor-phisms are sets. A category that is not small is said to be large. A category C islocally small if for any pair of objects A and B of C the class HomC(A,B) is a set.Many of the categories we will consider in this book (the categories of sets, groups,rings, modules over a ring, sheaves on a topological spaces, etc.) are locally small.

Given two categories C and D, a functor F : C→ D consists of the followingset of data:

1. a map Ob(C)→ Ob(D), A 7→ F (A);

2. a map HomC(A,B)→ HomD(F (A), F (B)) for any pair A,B ∈ Ob(C), suchthat F (f g) = F (f) F (g).

Let F and G be two functors from C to D. A morphism of functors θ : F → G

is a family of morphisms θA ∈ HomD(F (A), G(A)), one for each object A of C,such that the diagram

F (A)F (f) //

θA

F (B)

θB

G(A)

G(f) // G(B)

commutes for any morphism f ∈ HomC(A,B). A morphism of functors θ : F → G

is an isomorphism if and only if θA is an isomorphism for any object A.

A functor F : C→ D is said to be an equivalence of categories if there existsa functor G : D → C such that the composition FG is isomorphic to the identityfunctor IdC and the composition GF is isomorphic to the identity functor IdD.The functor G is said to be a quasi-inverse to F .

Given functors F : C → D and G : D → C, one says that G is left adjointto F (and that F is right adjoint to G) if there are functorial isomorphismsHomD(B,F (A)) ' HomC(G(B), A) for all objects A in C and B in D. If F isan equivalence of categories, its quasi-inverse is both right and left adjoint to F .Indeed, a functor F that admits a left adjoint G and a right adjoint H is anequivalence of categories if and only if G and H are isomorphic.

If C is a category, the opposite category C is the category whose objects arethe same as those of C and and whose morphisms are

HomC(A,B) = HomC(B,A) .

A functor F : C→ D is often called a contravariant functor from C to D.

A fundamental notion is that of representable functor.

A.2. Additive and Abelian categories 283

Definition A.1. A contravariant functor F : C→ Sets (where Sets is the categoryof sets) is said to be representable if there exists an object A in C such that F isisomorphic to the homomorphism functor HomC(•, A). 4

We recall that the homomorphism functor hA = HomC(•, A) is defined byletting hA(C) = HomC(C,A) for any object C, while for any morphism η : C → D

one defines hA(η) : HomC(D,A) → HomC(C,A) as the composition with η, i.e.,hA(η)(ϕ) = ϕ η.

Lemma A.2 (Yoneda’s lemma). Any isomorphism of functors

Φ: HomC(•, A) ≡ HomC(•, B)

is induced by an isomorphism A ' B.

Proof. For every object C we write ΦC : hA(C) → hB(C) for the induced map.We have a morphism ϕ = ΦA(IdA) : A→ B, and for every morphism η : C → A acommutative diagram

hA(A)ΦA //

hA(η)

hB(A)

hB(η)

hA(C)

ΦC // hB(C)

.

so that ΦA(η) = hB(η)(ϕ) = ϕη. This proves that Φ is induced by ϕ : A→ B. Asimilar argument proves that the inverse Φ−1 is induced by a morphism ψ : B → A.By functoriality, ϕψ induces the identity hB → hB and ϕψ induces the identityhA → hA, so that ψ and φ are isomorphisms, and one is the inverse of the other.

As a consequence of Yoneda’s Lemma A.2, a representable contravariantfunctor F is represented by an object which is unique up to isomorphisms.

A.2 Additive and Abelian categories

The most basic environment suitable to develop the machinery of homologicalalgebra is provided by additive categories.

Definition A.3. A category C is additive if the following conditions are satisfied:

1. for any A,B ∈ Ob(C), there is an Abelian group structure on HomC(A,B)such that all the composition maps

HomC(A,B)×HomC(B,C)→ HomC(A,C)

are bilinear;


2. there exists a zero object 0 such that HomC(0, 0) is the trivial group;

3. for any A,B ∈ Ob(C), there are an object Z ∈ Ob(C) and morphisms

AiA−→ Z

iB←− B , ApA←−− Z pB−−→ B

such that pA iA = IdA, pB iB = IdB , pA iB = 0, pB iA = 0 andpA iA + pB iB = IdZ (hence, Z is the direct sum and the direct productof A and B).

4

A functor F : C → D between two additive categories is said to be additiveif for any pair of morphisms f, g : A→ B one has F (f + g) = F (f) + F (g).

The kernel of a morphism f : A→ B is a morphism i : K → A such that forany object M , the sequence of Abelian groups 0→ Hom(M,K)→ Hom(M,A)→Hom(M,B) is exact. It easy to see that the kernel of a morphism, if it exists,is unique up to a unique isomorphism. Analogously, the cokernel of a morphismf : A → B is a morphism p : B → C such that for any for any object M , thesequence of Abelian groups 0 → Hom(C,M) → Hom(B,M) → Hom(A,M) isexact.

Assume that the morphism f : A → B has a cokernel p : B → coker f andthat p has a kernel. That kernel is called the image of f and is denoted by im f ;one has natural morphisms

im f = ker p u−→ Bp−→ coker f .

Suppose that f has a kernel i : ker f → A and that i has a cokernel. That cokernelis called the coimage of f and is denoted by coim f ; one has natural morphisms

ker f i−→ Aq−→ coker i = coim f .

Theorem A.4. Let C be an additive category. Assume that f : A → B is a mor-phism having a kernel, a cokernel, an image and a coimage. There exists a uniquemorphism f : coim f → im f such that the diagram

Af //

q

B

coim ff // im f

u

OO

is commutative.

A.2. Additive and Abelian categories 285

Definition A.5. Let k be a field. An additive category C is said to be k-linear iffor any objects A, B, the Abelian groups HomC(A,B) are k-vector spaces and thecomposition maps are k-bilinear. 4

An additive functor F : C→ D between two k-linear categories is said to be k-linear if for any morphism f : A→ B and any scalar λ ∈ k one has F (λf) = λF (f).

In C is a k-linear category, a contravariant k-linear functor F : C → Vectk(where Vectk is the category of k-vector spaces) is said to be representable if thereexists an object A in C such that F is isomorphic, as a k-linear functor, to thehomomorphism functor HomC(•, A). We also assume that all additive functors Fbetween two (k-linear) categories A and B are k-linear.

Definition A.6. An additive category is said to be Abelian if the following addi-tional conditions are satisfied:

1. any morphism has kernel and cokernel (consequently, any morphism hasimage and coimage);

2. for any morphism f : A → B, the morphism f : coim f → im f defined inTheorem A.4 is an isomorphism.

4

Remark A.7. A category is additive (resp. Abelian) if and only if its oppositecategory is additive (resp. Abelian). 4Example A.8. • The category of R-modules, where R is any ring, is Abelian.

• The category of free R-modules, where R is any commutative ring, is additivebut not Abelian.

• The category Mod(X) of sheaves of OX -modules on a ringed space (X,OX)is Abelian. This category will appear mostly in the case when X is an al-gebraic variety, i.e., a separated scheme of finite type over a field. Anothercase is the category of sheaves of Abelian groups on a topological space.

• The category Qco(X) of quasi-coherent sheaves of OX -modules on an alge-braic variety X is Abelian.

• The category Coh(X) of coherent sheaves of OX -modules on an algebraicvariety X is also Abelian.

4

Definition A.9. A sequence of morphisms . . .fn−2−−−→ An−1

fn−1−−−→ Anfn−→ An+1

fn+1−−−→. . . in an Abelian category A is said to be exact at An if ker fn = im fn−1. Thesequence is exact if it is exact at every term. 4


It is straightforward to show that a sequence 0 → A → B → C is exact ifand only if the sequence of Abelian groups 0 → Hom(M,A) → Hom(M,B) →Hom(M,C) is exact for any object M ∈ Ob(A).

Let F : A → B be an additive functor between two Abelian categories. Wesay that F is

• left exact if for any exact sequence 0 → A′ → A → A′′ → 0 in A, thesequence 0→ F (A′)→ F (A)→ F (A′′) is exact in B;

• right exact if for any exact sequence 0 → A′ → A → A′′ → 0 in A, thesequence F (A′)→ F (A)→ F (A′′)→ 0 is exact in B;

• exact if it is both right and left exact.

Though we shall not go into details, it is worth mentioning that we can defineArtinian, Noetherian and finite-length Abelian categories in terms of monomor-phisms, mimicking what one does for modules. However, to do so one does notneed all the conditions that Abelian categories fulfil. We now give for future usethe more general notion of quasi-Abelian category.

Let C be an additive category. A morphism f : A→ B in D having a kernel,a cokernel, an image and a coimage is strict if the natural morphism f : coim f →im f given by Theorem A.4 is an isomorphism. Note that an Abelian category isprecisely an additive category with kernels and cokernels (i.e., every morphism haskernel and cokernel) in which all morphisms are strict.

If C is an additive category with kernels and cokernels, it also has pull-backsand push-outs. The pull-back of a morphism f : A→ B by a morphism g : C → B

is the induced morphism g∗(f) : ker(f + g) → C where f + g : A ⊕ C → B isthe sum morphism; the push-out of f : A → B by a morphism h : A → C is theinduced morphism h∗(f) : C → im(f ×h) where f ×h : A→ B×C is the productmorphism.

Definition A.10. A quasi-Abelian category is an additive category with kernels andcokernels such that every pull-back of a strict epimorphism is a strict epimorphism,and every push-out of a strict monomorphism is a strict monomorphism. 4

An example is the category of torsion-free sheaves on a smooth projectivevariety.

A strict short exact sequence in a quasi-Abelian category is a sequence

0→ Ai−→ B

j−→ C → 0

in which i is the kernel of j and j is the cokernel of i (in particular, i is a strictmonomorphism and j a strict epimorphism).

A.3. Categories of complexes 287

Formulating the corresponding notions in terms of strict monomorphisms,it makes perfect sense to speak of Artinian, Noetherian and finite-length quasi-Abelian categories, as one does with Abelian categories.

A.3 Categories of complexes

Let A be an additive category. A complex (K•, dK•) in A is a sequence

· · · → Kn−1dn−1K•−−−→ Kn

dnK•−−→ Kn+1 → · · ·

where the Kn are objects in A and the morphisms dnK• are morphisms in A satis-fying the condition dn+1

K• dnK• = 0 for all n ∈ Z. We say that dK• is the differentialof the complex K•.

Definition A.11. The category of complexes C(A) is the category whose objectsare complexes (K•, dK•) in A and whose morphisms f : (K•, dK•)→ (L•, dL•) arecollections of morphisms fn : Kn → Ln, n ∈ Z, in A such that the diagrams

· · · // Kn−1 dn−1//

fn−1

Kn dn //

fn

Kn+1 dn+1//

fn+1

· · ·

· · · // Ln−1 dn−1// Ln dn // Ln+1 dn+1

// · · ·

are commutative. 4

Here and sometimes later on, when no ambiguity can arise, we omit thesubscripts in the symbols of the differentials.

Given two complexes K• and L•, their direct sum K•⊕L• is defined by setting(K⊕L)n = Kn⊕Ln and dnK•⊕L• = dnK• ⊕dnL• . If A has kernels and cokernels, andf : K• → L• is a morphism of complexes, its kernel is the complex ker f , such that(ker f)n = ker fn, endowed with the differential induced by dK• . In an analogousfashion one defines the cokernel of f . Hence, the following result holds true.

Proposition A.12. The category of complexes C(A) of an Abelian (resp. additive)category A is Abelian (resp. additive).

Remark A.13. The category A can be considered as a faithful subcategory of C(A).Indeed, any object A of A defines the complex A0 = A and An = 0 for n 6= 0,having the zero morphisms as differentials. 4Remark A.14. Assume that the Abelian category A has arbitrary direct sums,i.e., direct sums labeled by arbitrary sets (this is the case, for example, for thecategory of modules over a ring). Then, a complex K• in A can be regarded as a


graded object⊕

nKn and the differential dK• as a morphism⊕

nKn →⊕

nKnof degree 1. It follows that the morphisms of complexes from K• to L• form asubgroup of HomA(

⊕nKn,

⊕n Ln). Hence, the category of complexes C(A) is

itself an Abelian category with arbitrary direct sums. 4

Let K• and L• be complexes; for each n ∈ Z, we set

Hom(K•,L•)n =∏i

HomA(Ki,Li+n) .

These groups form a complex of Abelian groups

Hom•(K•,L•) =⊕n

Hom(K•,L•)n (A.1)

endowed with the differential given by

dn : Hom(K•,L•)n → Hom(K•,L•)n+1

f i 7→ di+nL• fi + (−1)n+1f i+1 diK• , .

(A.2)

Definition A.15. For any integer n, one defines the shift functor [n] : C(A)→ C(A)by lettingK[n]p = Kp+n with the differential dK•[n] = (−1)ndK• , while a morphismof complexes f : K• → L• is mapped to the morphism f [n] : K•[n] → L•[n] givenby f [n]p = fp+n. 4

The shift functor turns out to be additive and exact. Sometimes we shalldenote by τ the functor [1]. One has canonical isomorphisms

Hom•(K•,L•[n]) ' Hom•(K•,L•)[n] ' Hom•(K•[−n],L•) .

Definition A.16. The n-th cohomology object of a complex K• is the object

Hn(K•) = ker dn/ im dn−1 .

We say that Zn(K•) = ker dn is the n-cycle object of K•, and Bn(K•) = im dn−1

is the n-boundary object of K•. 4

The cohomology objects of a complex may be assembled into a complexH(K•) whose differentials are all set to zero.

A morphism of complexes f : K• → L• induces morphisms between cyclesand the boundaries, and passes to cohomology yielding morphisms

Hn(f) : Hn(K•)→ Hn(L•) ,

for every n. One has Hn(K•[m]) ' Hn+m(K•) and Hn(f [m]) ' Hn+m(f).


We say that a complex K• is acyclic or exact if H(K•) = 0; we also say thata morphism of complexes f : K• → L• is a quasi-isomorphism if H(f) : H(K•) →H(L•) is an isomorphism. The composition of two quasi-isomorphisms is a quasi-isomorphism.

We now introduce the important notion of homotopy equivalence, which willallow us to build a new category — the homotopy category — out of the categoryof complexes.

Let f : K• → L• a morphism of complexes. We say that f is homotopicto zero if there is a collection of morphisms hn : Kn → Ln−1 such that fn =hn+1 dnK•+dn−1

L• hn for every n. A complex K• is said to be homotopic to zero ifits identity morphism is homotopic to zero. Finally, two morphisms f, g : K• → L•are said to be homotopic if f − g is homotopic to zero.

It is clear that the sum of two morphisms homotopic to zero is homotopic tozero. Moreover, the composition f g is homotopic to zero whenever either f or gis homotopic to zero. Let us denote by Ht(K•,L•) the subgroup of the morphismsof complexes f : K• → L• which are homotopic to zero.

Definition A.17. The homotopy category K(A) is the category whose objects arethe objects of C(A) and whose morphisms are

HomK(A)(K•,L•) = HomC(A)(K•,L•)/Ht(K•,L•) .

4

From Equation (A.2) we see that the n-cycles of the complex of homomor-phisms Hom•(K•,L•) coincide with the morphisms of complexes K• → L•[n], whilethe n-boundaries coincide with morphisms homotopic to zero. Therefore,

Hn(Hom•(K•,L•)) = HomK(A)(K•,L•[n]) . (A.3)

A morphism of complexes f : K• → L• which is homotopic to zero inducesin cohomology the zero morphism, H(f) = 0; hence, two homotopic morphismsinduce the same morphism in cohomology. In particular, if a complex K• is homo-topic to zero, then it is acyclic.

We now define the cone of a morphism; this notion comes from classicalhomotopy theory, and is a way of overcoming the fact that the homotopy categoryK(A) does not have kernels and cokernels.

Definition A.18. The cone of a morphism of complexes f : K• → L• is the complexCone(f) such that Cone(f)n = Kn+1 ⊕ Ln and the differential is defined as

dnCone(f) =(−dn+1K• 0

fn+1 dnL•

)4


Although one has Cone(f)n = (K•[1])n ⊕ Ln for every n, Cone(f) is notisomorphic as a complex with the direct sum K•[1] ⊕ L•, because the differentialof the latter is the direct sum of the differentials of the summands. There arefunctorial morphisms

β : Cone(f)→ K•[1],

(k, l) 7→ k ,

α : L• → Cone(f)

l 7→ (0, l) ,

and an exact sequence of complexes 0 → L• → Cone(f) → K•[1] → 0. Let usconsider the sequence

K• f−→ L• α−→ Cone(f)β−→ K•[1] . (A.4)

The composition α f is homotopic to zero. If we consider the sequence (A.4) inthe homotopy category K(A), the composition of any two consecutive morphismsis zero. As we shall see in Section A.4.3, the sequence (A.4) will be the model forexact triangles in the homotopy and derived categories.

The following property is readily checked:

Proposition A.19. For every integer n there is an exact sequence of cohomologygroups

Hn(K•) Hn(f)−−−−→ Hn(L•) H

n(α)−−−−→ Hn(Cone(f))Hn(β)−−−−→ Hn(K•[1]) ' Hn+1(K•) .

Gathering all these exact sequences together we have the so-called cohomologylong exact sequence:

. . .Hn−1(β)−−−−−−→ Hn(K•) H

n(f)−−−−→ Hn(L•) Hn(α)−−−−→

Hn(Cone(f))Hn(β)−−−−→ Hn+1(K•) . . . (A.5)

Proposition A.19 tells us that the functorsHn : K(A)→ A are cohomological,in the following sense: if A and B are Abelian categories, an additive functorF : K(A) → B is cohomological if for every sequence K• f−→ L• α−→ Cone(f)

β−→K•[1] the sequence F (K•) F (f)−−−→ F (L•) α−→ F (Cone(f))

F (β)−−−→ F (K•)[1] is exact.

An important consequence of the cohomology long exact sequence (A.5) isthe following:

Corollary A.20. A morphism of complexes f : K• → L• is a quasi-isomorphism ifand only if Cone(f) is acyclic.

Cones are good substitutes for exact sequences of complexes, as the followingproposition shows (we omit the simple proof).


Proposition A.21. Let 0→ K• f−→ L• g−→ N • → 0 be an exact sequence of complexesin C(A), and let γ : Cone(f)→ N • be the morphism of complexes defined in degreen by

Kn+1 ⊕ Ln → Nn

(an+1, bn) 7→ g(bn) .

The morphism γ is a quasi-isomorphism.

Combining this with the cohomology long exact sequence (A.5), we obtainthe more usual cohomology sequence: there exist functorial morphisms

δn : Hn(N •)→ Hn+1(L•)

and an exact sequence

· · · δn−1

−−−→ Hn(L•)→ Hn(M•)→ Hn(N •) δn−→ Hn+1(L•)→

→ Hn+1(M•)→ Hn+1(N •) δn+1

−−−→ · · ·

Truncated complexes

Let K• be a complex. The truncations K•≤n and K•≥n are defined as the complexes

· · · → Kn−2 → Kn−1 → ker dn → 0 · · ·· · · → 0→ coker dn−1 → Kn+1 → Kn+2 → · · · ,

respectively. One has natural morphisms of complexes

K•≤n → K•, K• → K•≥n ,

and for any m ≤ n,

K•≤m → K•≤n, K•≥m → K•≥n .

A morphism of complexes K• → N • induces morphisms between the correspondingtruncations K•≤n → N •≤n and K•≥n → N •≥n.

Remark A.22. It is straightforward to check that

Hj(K•≤n) =

Hj(K•) for j ≤ n

0 for j > nHj(K•≥n) =

Hj(K•) for j ≥ n

0 for j < n

Thus, the truncation functors K• → K•≤n and K• → K•≥n preserve quasi-isomorphisms. Moreover, if we consider the natural morphism i : K•≤n−1 → K•≤n,the induced morphism Cone(i)→ Hn(K•)[−n] is a quasi-isomorphism. 4


A.3.1 Double complexes

A double complex K•• in an Abelian category A is a diagram

......

· · · // Kp,q+1

OO

d1 // Kp+1,q+1

OO

// . . .

· · · // Kp,qd2

OO

d1 // Kp+1,q

d2

OO

// . . .

...

OO

...

OO

(A.6)

where the Kp,q are objects of A, and one has

d21 = 0 , d2

2 = 0 and d1 d2 = d2 d1 .

A morphism of double complexes f : K•• → N •• is a collection of morphismsfp,q : Kp,q → N p,q commuting with the differentials d1 and d2.

A double complex K•• can be thought of as a complex of complexes in twodifferent ways. If we regard the columns Kp,• as complexes with differential d2, weget a complex

K•I = . . .d1−→ Kp,• d1−→ Kp+1,• d1−→ . . .

in the category C(A). Analogously, if we consider the rows K•,q as complexes withdifferential d1, we get a complex K•II in C(A) with differential d2. For any integern the cohomology Hn(K•I) is a complex with respect to d2, and Hn(K•II) is acomplex with respect to d1. We shall use the notation Hnd1

(K••) = Hn(K•I) andHnd2

(K••) = Hn(K•II).In many cases, it is useful to associate a complex with a double complex

K••. Let us assume that for any n the direct sum⊕

p+q=nKp,q exists. This is thecase for example when A admits infinite direct sums or when all antidiagonals indiagram (A.6) have only a finite number of nonzero terms. The simple complexS•(K••) associated to K•• consists of the objects

Sn(K••) =⊕p+q=n

Kp,q

and the differentials dn : Sn(K••)→ Sn+1(K••) such that dn = d1 + (−1)pd2 overKp,q.


Example A.23. We shall need to consider categories which admit tensor products.For an introduction to such categories we refer the reader, for instance, to ChapterVII of Mac Lane’s book [201]. Actually, for the concrete categories we shall dealwith, the tensor product structure is quite evident. So, letA be an Abelian categorywhich admits tensor products, and, in addition, infinite direct sums. If K• and L•are two complexes Kp⊗N q defines a double complex with differentials d1 = dK•⊗1,d2 = 1⊗ dL• . We can define the tensor product K• ⊗L• of the complexes K• andL• as the simple complex associated with this double complex. That is, one sets

(K• ⊗ L•)n =⊕p+q=n

(Kp ⊗ Lq)

equipping this complex with the differential which acts as dK•⊗Id+(−1)pId⊗dL•over Kp ⊗ Lq. If A has no infinite direct sums (as the category Coh(X)), thenK• ⊗L• is defined only if for every n there are only a finite number of summandsin⊕

p+q=n(Kp ⊗ Lq).The tensor product is compatible with the shift functor, i.e., one has canonical

isomorphismsK•[n]⊗ L• ' (K• ⊗ L•)[n] ' K• ⊗ L•[n] .

4Example A.24. The double complex of homomorphisms of two complexes K• andN • is defined by the objects

⊕p,q Hom(K−q,N p) with the differentials

d1 : Hom(K−q,N p)→ Hom(K−q,N p+1), f 7→ dN• fd2 : Hom(K−q,N p)→ Hom(K−q−1,N p), f 7→ (−1)q+1f dK• .

When the direct sum⊕

p+q=n Hom(K−q,N p) is isomorphic with the direct prod-uct

∏p+q=n Hom(K−q,N p), the associated simple complex is isomorphic with the

complex Hom•(K•,N •) as defined in (A.1). This happens, for instance, when K•is bounded above and N • is bounded below. 4Example A.25. A morphism of complexes f : K• → L• may be regarded as adouble complex with two columns. One sets K−1,• = K•, K0,• = L•, while theother columns are zero. The horizontal differential d1 : K−1,• → K0,• is f , and thevertical differential is given by the differentials of K• and L•. The simple complexassociated with this double complex is the cone of f . 4

Truncated double complexes

Let K•• be a double complex. The truncations K≤n,• and K≥n,• are defined as thedouble complexes

· · · → Kn−2,• → Kn−1,• → ker[d1 : Kn,• → Kn+1,•]→ 0 · · ·

· · · → 0→ coker[d1 : Kn−1,• → Kn,•]→ Kn+1,• → Kn+2,• → · · · ,


respectively. In a similar way by replacing rows by columns, one defines the trun-cated double complexes K•,≤n and K•,≥n.

Proposition A.21 and the content of Example A.25 imply the following result.

Proposition A.26. Consider the natural morphism i : S•(K≤n−1,•) → S•(K≤n,•).The induced morphism Cone(i) → Hnd1

(K••)[−n] is a quasi-isomorphism. There-fore, one has a long exact sequence

· · · → Hi(S•(K≤n−1,•))→ Hi(S•(K≤n,•))→→ Hid2

(Hnd1(K••)[−n])→ Hi+1(S•(K≤n−1,•))→ · · ·

An analogous result holds true for the truncated complex K•,≤n.

We say that a double complex K•• has bounded below antidiagonals (respec-tively, bounded above antidiagonals) if for each n one has Kp,n−p = 0 for p 0(respectively, Kn−q,q = 0 for q 0).

Proposition A.27. Let f : K•• → N •• be a morphism of double complexes. Assumethat both K•• and N •• have bounded below antidiagonals. If the induced morphism

Hd2(Hd1(K••))→ Hd2(Hd1(N ••))

is an isomorphism, the morphism S•(K••)→ S•(N ••) between the associated sim-ple complexes is a quasi-isomorphism. Analogously, assume that K•• and N •• havebounded above antidiagonals. If the induced morphism

Hd1(Hd2(K••))→ Hd1(Hd2(N ••))

is an isomorphism, the morphism S•(K••)→ S•(N ••) between the associated sim-ple complexes is a quasi-isomorphism.

Proof. Assume first that Kp,• = N p,• = 0 for p 0. The complexes S(K≤n−1,•)and S(N≤n−1,•) vanish for n 0, so the result holds for n small enough, and wemay use induction on n. We have a morphism of exact sequences

. . . // Hi(S•(K≤n−1,•)) //

Hi(S•(K≤n,•)) //

Hid2(Hnd1

(K••)[−n]) //

. . .

. . . // Hi(S•(N≤n−1,•)) // Hi(S•(N≤n,•)) // Hid2(Hnd1

(N ••)[−n]) // . . . .

Since by assumption the vertical morphism between the cohomology complexes isa quasi-isomorphism for every n, by Proposition A.26 the morphism S•(K≤n,•)→S•(N≤n,•) is a quasi-isomorphism for all n. This proves that S•(K••)→ S•(N ••) is

A.4. Derived categories 295

a quasi-isomorphism as well, since for each k one hasHk(S•(K••)) ' Hk(S•(K≤n,•))and Hk(S•(N •)) ' Hk(S•(N≤n,•)) when n is big enough.

To deal with the general case of double complexes with bounded below an-tidiagonals, it is enough to apply the previous argument to the induced morphismK≥n,• → N≥n,•. Indeed, for each k one has Hk(S•(K••)) ' Hk(S•(K≥n,•)) andHk(S•(N •)) ' Hk(S•(N≥n,•)) provided n is small enough. The case of doublecomplexes with bounded above antidiagonals is completely analogous.

Corollary A.28. Let K•• be a double complex with bounded below antidiagonals(respectively, bounded above antidiagonals). If there exists n such that Hid1

(K••) =0 for i 6= n (respectively, Hid2

(K••) = 0 for i 6= n), then Hi+n(S•(K••)) 'Hid2

(Hnd1(K••)) (respectively, Hi+n(S•(K••)) ' Hid1

(Hnd2(K••))).

Proof. By Proposition A.27, the morphismsHnd1(K••)[−n]← S•(K≤n,•)→ S•(K••)

are quasi-isomorphisms, so that

Hi(S•(K••)) ' Hi(Hnd1(K••)[−n]) ' Hi−nd2

(Hnd1(K••)) .

A.4 Derived categories

The basic idea behind the introduction of the derived category is to replacequasi-isomorphism with isomorphisms. The first step consists in identifying ho-motopic morphisms, thus moving from the category of complexes C(A) to thehomotopy category K(A). A second step consists in “localizing” by (classes of)quasi-isomorphisms. This localization is a fractional calculus for categories if wejust think of the composition of morphisms as a product. Recall that if one hasa ring A and we want to make the elements s in a part S of A invertible, sothat a fraction a/s makes sense, this can be done if S is a multiplicative system,namely, if it contains the unity and is closed under products. Then one can de-fine the localized ring S−1A whose elements are equivalence classes a/s of pairs(a, s) ∈ A × S where (a, s) ∼ (a′, s′) (or a/s = a′/s′) if there is t ∈ S such thatt(as′− a′s) = 0. Any element s ∈ S becomes invertible in the fractions ring S−1A

because s/1 · 1/s = 1.

A.4.1 The derived category of an Abelian category

The localization process can be also done for morphisms of complexes, since quasi-isomorphisms verify the conditions for being a nice set of denominators (or amultiplicative system as before), namely, the identity is a quasi-isomorphism and


the composition of two quasi-isomorphisms is a quasi-isomorphism. We now takea fraction as a diagram of morphisms of complexes

R•f

!!CCCCCCCCφ

K• L•

where φ is a quasi-isomorphism. We denote such a diagram by f/φ. A seconddiagram g/ψ

S•g

!!BBBBBBBBψ

K• L•

is said to be equivalent to the former, if there are quasi-isomorphisms R• ← T • →S• such that the diagram

T •

!!CCCCCCCC

R•

f**VVVVVVVVVVVVVVVVVVVVVVV

φ

S•

ψtthhhhhhhhhhhhhhhhhhhhhhh

g

!!BBBBBBBB

K• L•

is commutative in K(A). One can prove that equivalence of fractions is actuallyan equivalence relation using the following result.

Lemma A.29. Given a diagram

R•

g

M•

f // N •

in C(A), there are morphisms of complexes M• g′←− Z• f ′−→ R• such that thediagram

Z•f ′ //

g′

R•

g

M•

f // N •

is commutative in K(A). Moreover, f ′ (respectively, g′) is a quasi-isomorphism ifand only if f (respectively, g) is so.


Proof. Let us consider the morphism M• ⊕R• f−g−−−→ N • and set Z• = Cone(f −g)[−1]. The natural morphism Z• →M• ⊕R• induces morphisms g′ : Z• →M•

and f ′ : Z• → R•. The natural projection hn : Zn ' Mn ⊕Rn ⊕Nn−1 → Nn−1

defines a homotopy h between g f ′ and f g′. Finally, the commutative diagramof exact sequences

0 //M• //

f

M• ⊕R• //

f−g

R• //

0

0

0 // N • Id // N • // 0

induces an exact sequence of complexes

0→ Cone(f)[−1]→ Z• f ′−→ R• → 0 .

Hence, f ′ is a quasi-isomorphism if and only if Cone(f)[−1] is acyclic, and thisis equivalent by Corollary A.20 to f being a quasi-isomorphism. An analogousargument holds in the case of g and g′.

Definition A.30. The derived category D(A) of A is the category whose objectsare the objects of K(A) (that is, they are complexes of objects of A), and whosemorphisms are equivalence classes [f/φ] of diagrams. 4

To make full sense of this definition we need to specify how to composemorphisms. This can be done thanks to Lemma A.29. Given two morphisms [f/φ]and [g/ψ] in D(A), corresponding to diagrams

R•f

!!BBBBBBBBφ

||||||||S•

g

""EEEEEEEEψ

~~||||||||

K• L• M• ,

their composition is defined through the diagram

T •ψ′

f ′

!!CCCCCCCC

R•f

!!CCCCCCCCφ

S•

g

""EEEEEEEEψ

K• L• M• .

Hence, we set [g/ψ] [f/φ] = [(g f ′)/(φ ψ′)], which makes sense because theabove construction is independent of the representatives.


A result similar to Lemma A.29 holds true by inverting all the arrows. Adiagram

M• //

N •

R•

in C(A) can be completed to a diagram

M• //

N •

R• // Z•

which is commutative in K(A). Moreover, each arrow of this diagram is a quasi-isomorphism if and only if its parallel arrow is so. Putting altogether, a morphismfrom K• to L• in the derived category is also defined by an equivalence class [ψ\g]of diagrams

K•g

!!CCCCCCCC L•ψ

R•

where ψ a quasi-isomorphism. The composition is defined in a completely analo-gous way.

By a slight abuse of notation, we shall simply denote by f/φ : K• → L• themorphism in the derived category defined by the equivalence class [f/φ] and byφ\f : K• → L• the morphism defined by the equivalence class [φ\f ].

Useful descriptions of morphisms in the derived category are provided by thefollowing isomorphisms:

HomD(A)(K•,M•) ' lim−→I•

HomK(A)(K•, I•) ' lim−→P•

HomK(A)(P•,M•) , (A.7)

where the first limit runs over all the quasi-isomorphisms of complexesM• → I•,while the second runs over all the quasi-isomorphisms of complexes P• → K•.

A morphism f : K• → L• in C(A) defines a morphism f/IdK• : K• → L•in the derived category, which we shall denote simply by f . Moreover, if f ishomotopic to zero, f/IdK• = 0. Hence, we have a functor K(A)→ D(A).

Proposition A.31. The derived category D(A) is an additive category and the func-tor K(A)→ D(A) is additive.

Proof. Two morphisms f/φ : K• → M• and g/ψ : L• → M• yield a morphism(f + g)/(φ⊕ ψ) : K• ⊕ L• →M•. Conversely, a morphism K• ⊕ L• →M• in the


derived category, being represented by a morphism of complexes f : K•⊕L• → R•and a quasi-isomorphism φ : M• → R•, defines morphisms φ\fK• : K• → R• andφ\fL• : L• →M• in the derived category, where fK• : K• → R• and fL• : L• → R•are the morphisms induced by f . This shows that K•⊕L• is the direct sum in thederived category. In a similar way, one proves that K•⊕L• is the direct product inthe derived category. The sum of two morphisms f/φ : K• → L• and g/ψ : K• → L•is defined as the composition of the diagonal morphism K• → K• ⊕ K• and thedirect sum (f + g)/(φ⊕ ψ) : K• ⊕K• → L•.

The additivity of the functor K(A)→ D(A) is easily checked.

Definition A.32. Two complexes K• and L• are quasi-isomorphic if there are acomplex Z• and quasi-isomorphisms K• ← Z• → L•. 4

It follows from Lemma A.29 that the notion of quasi-isomorphism induces anequivalence relation between complexes. One can also prove that two complexesK• and L• are quasi-isomorphic if and only if there are a complex Z• and quasi-isomorphisms K• → Z• ← L•.

We now have the result we were looking for:

Proposition A.33. A morphism of complexes f : K• → L• is a quasi-isomorphismif and only if the induced morphism in the derived category is an isomorphism.Moreover, two complexes are quasi-isomorphic if and only if they are isomorphicin the derived category.

Proof. If f is a quasi-isomorphism, we can define a morphism IdK•/f : L• → K•in the derived category, which is precisely the inverse of f . Conversely, if f is anisomorphism in the derived category and g/ψ is its inverse, then H(g/ψ) is theinverse of H(f). The second statement follows straightforwardly.

The derived category can be also defined by means of a universal property.

Proposition A.34. Let C be an additive category. An additive functor F : K(A)→C factors through an additive functor D(A) → C if and only if it maps quasi-isomorphisms to isomorphisms. If B is an Abelian category, an additive functorG : K(A)→ K(B) mapping quasi-isomorphisms into quasi-isomorphisms inducesan additive functor G : D(A)→ D(B) such that the diagram

K(A) G //

K(B)

D(A) G // D(B)

is commutative.


Proof. If F factors through D(A), it maps quasi-isomorphisms to isomorphismsby Proposition A.33. Conversely, if F maps quasi-isomorphisms to isomorphisms,one defines a functor D(F ) : D(A) → C by letting D(F )(M•) = F (M•) for anyobject M• and D(F )(f/φ) = F (f) F (φ)−1 for any morphism f/φ : K• → M•

in D(A). By applying this to the composition F : K(A) → K(B) G−→ D(B), thesecond statement follows.

Since a quasi-isomorphism of complexes induces a quasi-isomorphism be-tween the truncated complexes (Remark A.22), the truncations functors pass tothe derived category.

Corollary A.35. There exist additive functors

(−)≥n : D(A)→ D(A) , (−)≤n : D(A)→ D(A) .

A.4.2 Other derived categories

We can also build derived categories out of some subcategories of C(A), as long asall the operations we have done so far can be reproduced; namely, we need to con-struct the corresponding homotopy categories, and localize by quasi-isomorphims.In particular, we must define the cone of a morphism inside the new category(cf. for instance Lemma A.29, whose proof requires the cone construction).

The most natural examples are the following.

Example A.36. Let C+(A) be the category of bounded below complexes in A, thatis, complexes K• for which there is n0 such that Kn = 0 for all n ≤ n0. We candefine the homotopy category K+(A) and a derived category D+(A) by followingan analogous procedure as for arbitrary complexes. Due to Proposition A.34, thenatural functor K+(A) → D(A) induces a functor γ : D+(A) → D(A). This isfully faithful and its essential image is the faithful subcategory of D(A) consistingof complexes in A with bounded below cohomology. (The essential image of thefunctor γ is the subcategory of the objects which are isomorphic to objects ofthe form γ(K•) for some K• in D+(A)). In a similar way, the categories C−(A) ofbounded above complexes (i.e., complexes for which there is n0 such that Kn = 0 forall n ≥ n0) and Cb(A)) of bounded (on both sides) complexes, give rise to derivedcategories D−(A) and Db(A), which are characterized as faithful subcategories ofD(A) as above. 4

Example A.37. Let A′ be a thick Abelian subcategory of A, that is, any extensionin A of two objects of A′ is also in A′. If CA′(A) is the category of complexes whosecohomology objects are in A′, we can construct its homotopy category KA′(A) and


its derived category DA′(A). The functor KA′(A)→ D(A) induces by PropositionA.34 a functor DA′(A) → D(A), which is fully faithful; its essential image is thesubcategory of D(A) whose objects are all complexes with cohomology objects inA′. 4Example A.38. Combining the two procedures, we also have the homotopy cat-egories K+

A′(A), K−A′(A) and KbA′(A) of complexes bounded below, above and on

both sides, respectively, and whose cohomology objects are in the subcategoryA′ of A. We also have the corresponding derived categories D+

A′(A), D−A′(A) andDb

A′(A). 4

We have special notations for the Abelian categories we are most interestedin.

• If A is the category of modules over a commutative ring A, we use thenotations D(A), D+(A), D−(A), and Db(A).

• If A = Mod(X) is the category of sheaves of OX -modules on an alge-braic variety X and A′ = Qco(X) is the category of quasi-coherent sheavesof OX -modules on X, the derived category DA′(A) of complexes of OX -modules with quasi-coherent cohomology sheaves is denoted Dqc(Mod(X)).In a similar way we have the categories D+

qc(Mod(X)), D−qc(Mod(X)) andDbqc(Mod(X)).

• If A = Mod(X) as above and A′ = Coh(X) is the category of coherentsheaves of OX -modules on X, the derived category DA′(A) of complexesof OX -modules with coherent cohomology sheaves is denoted Dc(Mod(X)).One can also introduce the derived categories D+

c (Mod(X)), D−c (Mod(X))and Db

c(Mod(X)).

• If A = Qco(X) and A′ = Coh(X), the derived category DA′(A) of complexesof quasi-coherent OX -modules with coherent cohomology sheaves will bedenoted by D(X) for simplicity. Also, the corresponding derived categoriesof bounded below, bounded above and bounded complexes are denoted byD+(X), D−(X), and Db(X).

Let us write ? for any of the symbols +, −, b or for no symbol at all. Since thenatural functors K?(A′)→ D(A) map quasi-isomorphisms to isomorphisms, theyyield functors D?(A′) → D?

A′(A), which in general may fail to be equivalences ofcategories.

However, if A′ has enough injectives in A, that is, if for every object K of A′

there is an immersion 0→ K → I where I is an object of A′ which is injective in A,one can prove that every bounded below complex K• in K+(A) whose cohomologyobjects are in A′ admits a quasi-isomorphism

K• → I• ,


where I• is a complex of objects of A′ which are injectives in A (cf. [139]). Thenone has:

Proposition A.39. If A′ has enough A-injectives, the functor

D+(A′)→ D+(A)

is fully faithful and induces an equivalence of categories D+(A′) ' D+A′(A).

Proof. Let K• and L• be objects of D+(A′). A morphism K• → L• in D+(A) canbe represented by a diagram

K•g

!!CCCCCCCC L•ψ

R•

where R• is in D+(A) and ψ a quasi-isomorphism. Then R• is in D+A′(A), so that,

as have seen, there is a quasi-isomorphism γ : R• → I• with I• ∈ K+(A′). Thus,K• → L• is also represented by the diagram

K•γg

!!BBBBBBBB L•γψ

||||||||

I•,

which defines a morphismK• → L• inD+(A′). It follows that HomD+(A′)(K•,L•) =HomD+(A)(K•,L•). Moreover, since any object of D+

A′(A) is quasi-isomorphic to acomplex in D+(A′), the essential image of D+(A′)→ D+(A) is D+

A′(A).

If for any complex K• in K−A′(A) there is complex L• in K−(A′) and a quasi-isomorphism L• → K•, proceeding as above (but using the representation of mor-phisms in the derived category given by Definition A.30), we may check that thereis an equivalence of categories

D−(A′) ' D−A′(A) , (A.8)

and similarly for bounded complexes. For the derived categories associated withan algebraic variety X, one has the following result.

Proposition A.40. Let X be an algebraic variety. There is an equivalence of cate-gories

D+(Qco(X)) ' D+qc(Mod(X)) .


Proof. This follows from Proposition A.39, as every quasi-coherent sheaf K on analgebraic variety can be embedded as a subsheaf of an injective quasi-coherentsheaf.

Corollary A.41. If X is an algebraic variety, one has an equivalence of categories

Db(Coh(X)) ' Db(X) ' Dbc(Mod(X)) .

Proof. In order to prove the first isomorphism, the only nontrivial thing to show isthat the natural functor Db(Coh(X)) → Db(X) is essentially surjective, i.e., thatevery bounded complex E• of quasi-coherent sheaves with coherent cohomologysheaves is quasi-isomorphic to a bounded complex G• of coherent sheaves. A quasi-isomorphism G• → E•, where G• is a bounded complex of coherent sheaves, isconstructed by standard techniques (as in [229, Lemma II.1]) by exploiting thefollowing fact: given a surjection of quasi-coherent sheaves E → H → 0 whereH• is coherent, there exists a coherent subsheaf G of E such that the compositionE → H is surjective as well (cf. [141, Exercise II.5.15]. The second isomorphismfollows from Proposition A.40.

Definition A.42. A complex F• of OX -modules is of finite homological dimension,or a perfect complex, if it is locally quasi-isomorphic to a bounded complex oflocally free sheaves of finite rank. That is, every point x has an open neighborhoodU such that F•|U is quasi-isomorphic to a complex 0 → En → · · · → En+m → 0of locally free OU -modules of finite rank. 4

When X is smooth, every bounded complex of OX -modules with coherentcohomology sheaves is a perfect complex.

A.4.3 Triangles and triangulated categories

The notion of triangle is a replacement of that of exact sequence which is wellsuited for derived categories, which are not Abelian. Indeed, the derived categoryof an Abelian category has a natural structure of triangulated category.

Definition A.43. A triangle in D(A) is a sequence of morphisms

A• u−→ B• v−→ C• w−→ A•[1] .

4

A triangle is also written in the form

A• u // B•

v~~||||||||

C•w

aaBB

BB

,


where the dashed arrow stands for a morphism C• → A•[1]. For any morphism ofcomplexes f : K• → L• one has a triangle

K• f−→ L• → Cone(f)→ K•[1] .

A morphism of triangles is a commutative diagram

A• u //

f

B• v //

g

C• w //

h

A•[1]

f [1]

A′• u′ // B′• v′ // C′• w′ // A′•[1]

.

A triangle is called exact if it is isomorphic to a triangle of the type K• f−→ L• →Cone(f)→ K•[1]. For example, if 0→ K• i−→ L• →M• → 0 is an exact sequenceof complexes, then

K• i−→ L• →M• → K•[1]

is an exact triangle, where M• → K•[1] is the morphism in D(A) given by thediagram

Cone(i)

$$IIIIIIIII

vvvvvvvvv

M• K•[1]

.

A triangle in D(A) induces a long sequence in cohomology

· · · → Hi(A•) Hi(u)−−−−→ Hi(B•) Hi(v)−−−−→ Hi(C•) Hi(w)−−−−→

Hi+1(A•) Hi+1(u)−−−−−→ Hi+1(B•) Hi+1(v)−−−−−→ Hi+1(C•) Hi+1(w)−−−−−→ · · · .

If the triangle is exact, this long sequence is exact.

Another interesting example of an exact triangle is constructed out of thetruncation functors: since for any integer n the natural morphism i : K•≤n−1 →K•≤n induces a quasi-isomorphism Cone(i) → Hn(K•)[−n], we have an exacttriangle in Db(A):

K•≤n−1 → K•≤n → Hn(K•)[−n]→ K•≤n−1[1] .

We enumerate here the principal properties of exact triangles.

Proposition A.44. The collection of exact triangles in D(A) satisfies the followingproperties:


(TR0) A triangle isomorphic to an exact triangle is exact.

(TR1) For any object A• in D(A), the triangle A• Id−→ A• → 0→ A•[1] is exact.

• → B• in D(A) can be embedded into an exact triangle

A• f−→ B• → C• → A[1].

(TR3) A triangle A• u−→ B• v−→ C• w−→ A•[1] is exact if and only if B• v−→ C• w−→A•[1]

u[1]−−→ B•[1] is an exact triangle.

(TR4) Given two exact triangles A• u−→ B• v−→ C• w−→ A•[1] and A′• u′−→ B′• v′−→C′• w′−→ A′•[1], a commutative diagram

A• u //

f

B•

g

A′• u′ // B′•

can be embedded into a morphism between the two triangles (not necessarilyunique).

(TR5) (Octahedral axiom). Given exact triangles

A• u // B• // C′• // A•[1],

B• v // C• // A′• // B•[1],

A• w // C• // B′• // A•[1],

there exists an exact triangle C′• → B′• → A′• → C′•[1] such that the follow-ing diagram is commutative:

A• u //

Id

B• //

v

C′• //

A•[1]

Id

A• w //

u

C• //

Id

B′• //

A•[1]

u[1]

B• v //

C• //

A′• //

Id

B•[1]

C′• // B′• // A′• // C′•[1] .

(TR2) Any morphism f :A


These properties of exact triangles are the model for the notion of triangu-lated category, which we proceed to define.

Let A be an additive category with an automorphism τ : A → A, which wecall the translation or shift functor. We use the notation a[1] = τ(a) for any objecta in A. A triangle in A is a sequence of morphisms

au−→ b

v−→ cw−→ a[1] .

A morphism of triangles is a commutative diagram

au //

f

bv //

g

cw //

h

a[1]

f [1]

a′

u′ // b′v′ // c′

w′ // a′[1] .

Definition A.45. An additive category A is a triangulated category if it has ashift functor τ : A → A as above, and a collection of triangles, called the exactor distinguished triangles of A, which fulfil the axioms TR0 to TR5 listed inProposition A.44. 4

It is then clear from Proposition A.44 that the various derived categoriesD?(A) associated with an Abelian category A are triangulated categories. Thehomotopy categories K?(A) are triangulated as well, provided one decrees that atriangle is exact when it is isomorphic, in the obvious sense, to the triangle definedby the cone of a morphism (cf. Eq. (A.4)).

A full subcategory A′ of A is a triangulated subcategory if for any object aof A′, its shift a[1] is also an object of A′, and for any exact triangle in A suchthat two of its vertices are in A′, the third vertex is in A′ as well.

Definition A.46. Let A and B be triangulated categories. A covariant functorF : A→ B is said to be exact if it satisfies:

1. F is additive and commutes with the shift functor, F (a[1]) ' F (a)[1].

2. For any exact triangle a→ b→ c→ a[1] in A, the triangle in B

F (a)→ F (b)→ F (c)→ F (a[1]) ' F (a)[1]

is exact.

4

In particular, the shift functor is an exact functor.

The notion of cohomological functor that we have already met in this ap-pendix makes sense in the general context of triangulated categories.


Definition A.47. Let A be a triangulated category. A functor H from A to thecategory of Abelian groups is said to be cohomological if it maps exact trianglesto long exact sequences, that is, for every exact triangle a → b → c → a[1] thesequence

· · · → H(a[i])→ H(b[i])→ H(c[i])→ H(a[i+ 1])→ . . .

is exact. One analogously defines the notion of contravariant cohomological functorby reversing the arrows in the last sequence. 4

A.4.4 Differential graded categories

Most triangulated categories that appear in algebra and in algebraic geometry canbe obtained as “homotopy categories” associated to certain more highly structuredcategories: the differential graded categories. The idea of enhancing triangulatedcategories was originally proposed by Bondal and Kapranov [46]. In this section,we give a brief account of the definition and basic properties of dg-categories,following quite closely the excellent review paper [178].

Definition A.48. A k-linear category C is said to be a differential graded category(or dg-category for short) if the morphism spaces are differential graded k-vectorspaces (i.e., complexes of k-vector spaces) and the composition maps

HomC(A,B)⊗HomC(B,C)→ HomA(A,C)

are morphisms of differential graded k-vector spaces. 4

The opposite category C of a dg-category C is also a dg-category, if onedefines the composition of two morphisms f ∈ HomC(A,B)p = HomC(B,A)p

and g ∈ HomC(B,C)q = HomC(C,B)q as (−1)pqf g.

A functor F : C→ D between two dg-categories is said to be a dg-functor ifit induces, for any A,B ∈ Ob(C), morphisms of differential graded k-vector spacesF (A,B) : HomC(B,C)→ HomD(F (A), F (C)) which are compatible with the com-position maps and the units. One can also define the notions of dg-subcategory andfull dg-subcategory in the obvious way.

A graded category is a dg-category such that for any A,B ∈ Ob(C), thedifferentials of the complex HomC(A,B) are zero. We denote by Cgr the gradedcategory naturally associated to the dg-category C.Example A.49. 1. Any dg-algebra A is nothing but a dg-category with one object.Thus dg-categories can be thought of as “dg-algebras with many objects.”

2. The category of complexes C(A) of a k-linear Abelian category A can bemade into a dg-category, Cdg(A), by setting

HomCdg(A)(K•,L•) = Hom•(K•,L•) =⊕n

Hom(K•,L•)n (A.9)


(cf. (A.1)) with the differential given by (A.2).

3. The subcategories C+(A), C−(A) and Cb(A) carry natural structures offull dg-subcategories of Cdg(A). 4

The homotopy category H0(C) has the same objects as C, while the morphismspaces are the zero-cohomology groups of the complex HomC(A,B), namely

HomH0(C)(A,B) = H0(HomC(A,B)) .

A dg-functor F : C → D between two dg-categories induces a homotopy functorH0(F ) : H0(C) → H0(D). We can also define the category Z0(C) with the sameobjects as C and morphism spaces given by the zero-cycles:

HomZ0(C)(A,B) = Z0(HomC(A,B)) .

A dg-functor F : C→ D is a quasi-equivalence if the following conditions aresatisfied:

1. F (A,B) is a quasi-isomorphism for all objects A,B ∈ Ob(C);

2. H0(F ) : H0(C)→ H0(D) is an equivalence.

The category of dg-categories

One important feature of dg-categories is that they form a dg-category as well(provided one takes the precaution to deal only with small categories).

Definition A.50. The category of small dg-categories is the category dgcatk withthe dg-categories as objects and the dg-functors as morphisms. 4

The dg-category dgcatk has an initial object, the empty category, and a finalobject, the dg-category with one object whose endomorphism ring is zero. Thetensor product of two dg-categories C, D is the category C ⊗ D whose objectsare pairs (A,B), where A is an object of C and B is an object of D, and whosemorphism spaces are the graded tensor products

HomC⊗D((A,B), (A′, B′)) = HomC(A,A′)⊗HomD(B,B′) .

If F : C → D and G : C → D are dg-functors, one defines the complex ofgraded homomorphisms Hom(F,G) = ⊕nHomn(F,G), where Homn(F,G) is thefamily of morphisms φA ∈ Homn

D(F (A), G(A)) such that G(f) φA = φB F (f)for all morphisms f ∈ HomC(A,B). The differential is induced by the differentialof Homn

D(F (A), G(A)). Notice that the set of morphisms F → G is Z0Hom(F,G).


One defines the dg-category Hom(C,D) whose objects are the dg-functorsfrom C to D and whose graded spaces of morphisms are Hom(F,G).

The category dgcatk, equipped with the tensor product, becomes a symmetrictensor category admitting an internal Hom-functor, namely,

Hom(B⊗ C,D) = Hom(B,Hom(C,D))

for any B,C,D ∈ Ob(dgcatk).

By relying on the techniques developed by Drinfeld in [105], Tabuada provedthe following fundamental result.

Theorem A.51. [274] The category dgcatk admits a structure of cofibrantly gener-ated model category, whose weak equivalences are the quasi-equivalences.

Quillen’s notion of model category (introduced in [256]) provides a generalsetting where it is possible to develop the basic machinery of homotopy theories.One denotes by Ho(dgcatk) the “homotopy category” associated to dgcatk, whichcan be realized as the localization of dgcatk with respect to quasi-equivalences(cf. [106, § 5, 6]).

The derived category of a dg-category

If A is a differential graded algebra, any right differential graded module M overA can be viewed as a dg-functor A → Cdg(k), where A is the category with oneobject associated to A and Cdg(k) is the dg-category of complexes of k-vectorspaces. In the same vein, a right dg-module over a small dg-category C is definedas a dg-functor M : C → Cdg(k) and a morphism of right dg-modules is definedas a morphism of dg-functors. Equivalently, a right dg-C-module M is specifiedby giving for each object C of C a complex M(C) of k-vector spaces, and for eachpair of objects C and D, a morphism of complexes M(D)⊗HomC(C,D)→M(C)in a compatible way with compositions and units. The shift M [n] (n ∈ Z) of aright dg-C-module is defined by M [n](C) = M(C)[n] for any object C in C.

The category C(C) of right dg-C-modules is an Abelian category. A mor-phism M → N of dg-C-modules is an epimorphism (resp. a monomorphism, aquasi-isomorphism) if for every object C in C the morphism of complexes of k-vector spaces M(C)→ N(C) is an epimorphism (resp. a monomorphism, a quasi-isomorphism).

Right dg-modules can be used to construct the derived category of a dg-category C. We define the dg-category of complexes of right dg-C-modules as thedg-category

Cdg(C) = Hom(C,Cdg(k)) ,


whose objects are the right dg-C-modules and whose morphism spaces are thecomplexes of graded homomorphisms of dg-functors. Notice that there is an equiv-alence of categories C(C) ∼→ Z0(Cdg(C)).

The homotopy category of the category Cdg(C) is denoted by

H(C) = H0(Cdg(C)) .

It is a triangulated category [177, (2.2)].

It is worth mentioning that there is a dg-functor C → Cdg(C), given byA 7→ hA, where hA is the right dg-C-module defined by hA(B) = HomC(A,B)).It is fully faithful, that is, one has HomC(A,B) = HomCdg(C)(hA, hB) (Yoneda’sformula). A right dg-C-module M is said to be representable (resp. representableup to homotopy) if it is isomorphic in Cdg(C) (resp. in H(C)) to hA for some objectA ∈ Ob(C).

In complete analogy to the procedure used to introduce the usual notionof derived category for an Abelian category (cf. Definition A.30), one gives thefollowing definition.

Definition A.52. The derived category D(C) of the dg-category C is the localizationof H(C) with respect to the class of quasi-isomorphisms. 4

One says that a right dg-C-module M is quasi-representable if it is isomorphicin D(C) to hA for some A ∈ Ob(C).

Remark A.53. When C is an Abelian category, so that HomC(A,B) = Hom0C(A,B)

for each pair of objects A, B in C, it turns out that D(C) is the usual derivedcategory. 4

Toen’s results

Let us recall the notions of cofibrant and fibrant dg-modules, referring the reader to[177] and [284] for further details. A dg-C-module P is cofibrant if for every surjec-tive quasi-isomorphismM → N of dg-C-modules, the morphism HomC(C)(P,M)→HomC(C)(P,N) is an epimorphism. A dg-C-module I is fibrant if for every injec-tive quasi-isomorphism M → N of dg-C-modules, the morphism HomC(C)(N, I)→HomC(C)(M, I) is an epimorphism.

It can be shown that for each C-module M , there are quasi-isomorphismspM →M and M → iM where pM is cofibrant and iM is fibrant [177]. Moreover,pM and iM are unique up to homotopy, and the natural functor H(C) → D(C)admits a fully faithful left adjoint given by M 7→ pM and a fully faithful rightadjoint given by M 7→ iM .

When C is the dg-category associated to a k-algebra A and M is a right


A-module, pM → M is a projective resolution and M → iM is an injectiveresolution.

Definition A.54. The dg-derived category of a dg-category C is the full dg-subcate-gory Ddg(C) of Cdg(C) whose objects are all the cofibrant right dg C-modules. 4

Remark A.55. The previous definition is taken from [178], in the spirit of [284]. Fora different, but equivalent definition of the categoryDdg(C) based on a constructionwhich generalizes Verdier’s quotient to the differential graded setting, see [177] and[105]. 4

The homotopy category H0(Ddg(C)) is a triangulated subcategory of H(C) =H0(Cdg(C)) and the functor Cdg(C)→ H(C)→ D(C) induces an exact equivalenceof triangulated categories

H0(Ddg(C)) ∼→ D(C) .

Let us consider two dg-categories C, D. We can form the dg-category of C-D-bimodules (i.e., right dg-(C ⊗D)-modules, according to our notation). Givenan object A in C, one defines the natural dg-functor iA : D → C ⊗D given byB 7→ A⊗B. A C-D-bimodule N is called quasi-representable if for any A ∈ Ob(C)the right dg-D-module i∗A(N) is quasi-representable (cf. [284, Def. 4.1]). Quasi-representable C-D-bimodules are also called quasi-functors because they inducefunctors H(C)→ H(D).

As we have noticed, the category dgcatk is a tensor category admitting aninternal Hom-functor Hom. The tensor product ⊗ induces a tensor product, de-

notedL⊗ , on the category Ho(dgcatk), since the functor C ⊗ • preserves weak

equivalences when C is cofibrant. On the other hand, even in the case when C iscofibrant, the functor Hom(C, •) does not preserve weak equivalences. In spite ofthat, the following result can be proved.

Theorem A.56. [284, Theorem 6.1] The monoidal category (Ho(dgcatk),L⊗ ) ad-

mits an internal Hom-functor RHom. For any dg-categories C, D, the dg-categoryRHom(C,D) is naturally isomorphic in Ho(dgcatk) to the dg-category of cofibrantright quasi-representable C-D-bimodules, i.e., to the category of quasi-functorsfrom C to D.

Let us now take an algebraic variety X and the Abelian category Qco(X) ofquasi-coherent sheaves on it. We can form the dg-category Cdg(Qco(X)) (cf. Ex-ample A.49), and construct the dg-derived category Ddg(Cdg(Qco(X))), which wedenote simply by Ddg(X). As we already remarked, one has H0(Ddg(X)) ' D(X).The following result is particularly relevant to the purposes of this book (see Sec-tion 2.4).


Theorem A.57. [284, Theorem 8.9] Let X and Y be algebraic varieties over k.There is a natural isomorphism in Ho(dgcatk)

Ddg(X × Y ) ' RHomc(Ddg(X), Ddg(Y )) ,

where RHomc denotes the full subcategory of RHom consisting of coproduct pre-serving quasi-functors. In particular, the set of isomorphism classes of objects inthe derived category D(X ×k Y ) is isomorphic to the set of direct sum preservingmorphisms between Ddg(X) and Ddg(Y ) in Ho(dgcatk).

A more easily handled version of the previous theorem can be proved in the casewhen X and Y are smooth projective varieties. Let us denote by parfdg(X) the fullsub-dg-category of Ddg(X) whose objects are the perfect complexes (cf. DefinitionA.42). Then, one has [284, Theorem 8.15]

parfdg(X ×k Y ) ' RHom(parfdg(X),parfdg(Y )) . (A.10)

A.4.5 Derived functors

We know that the cohomology groups of a sheaf F on an algebraic variety X arethe cohomology objects of the complex of global sections Γ(X, I•) of a resolutionI• of F by injective sheaves (two different injective resolutions I• and J • give riseto the same cohomology groups). Indeed, the complexes Γ(X, I•) and Γ(X,J •) arequasi-isomorphic, so that they define isomorphic objects in the derived categoryof the category of Abelian groups. We can then associate with F a single objectRΓ(X,F) = Γ(X, I•) ' Γ(X,J •).

This is the procedure we mimic to define derived functors on the derivedcategory. To simplify the construction we strengthen the notion of “having enoughinjectives”: this will mean that there is a functor I : A→ A (where A is an Abeliancategory) such that I(M) is injective for any object M∈ A, and that there is animmersion 0 → M → I(M) which depends functorially on M. It can be easilychecked that if X is an algebraic variety, the categories Mod(X) and Qco(X) haveenough injectives also in this stronger sense.

It follows that any object M in A has an injective resolution

M→ I0(M)→ I1(M)→ . . .

which is functorial in M. If M• is a bounded below complex, we consider thedouble complex I•(M•) obtained by associating to any term in the complex M•

its injective resolution. The associated simple complex is denoted by I(M•). Thisdefines a functor I : K+(A)→ K+(A). Moreover, by Proposition A.27, the naturalmorphism

M• → I(M•)


is a quasi-isomorphism.

Let now B be another Abelian category and F : A→ B a left exact functor.Then F induces a functor RF : K+(A) → D+(B) by RF (M•) = F (I(M•)). IfJ • is an acyclic complex of injective objects, then F (J •) is acyclic, because J •splits. Since a morphism of complexes is a quasi-isomorphism if and only if itscone is acyclic (cf. Corollary A.20), we deduce that RF maps quasi-isomorphismsto isomorphisms, and then by Proposition A.34 it yields a functor

RF : D+(A)→ D+(B) ,

which is the right derived functor of F .

We shall denote RiF (M•) = Hi(RF (M•)). The restriction of the functorRiF : D+(A)→ B to A is the “classical” i-th right derived functor of F .

The right derived functor RF is exact, that is, it maps exact triangles toexact triangles. In particular, an exact triangle in D+(A)

M′• →M• →M′′• →M′•[1]

induces a long exact sequence

· · · → RiF (M′•)→ RiF (M•)→ RiF (M′′•)→Ri+1F (M′•)→ Ri+1F (M•)→ Ri+1F (M′′•)→ · · · .

For any bounded below complex M• there is a natural morphism

F (M•)→ RF (M•)

in the derived category. The complexM• is said to be F -acyclic if this morphismis an isomorphism, that is, M• ' RF (M•) in D+(B).

One can develop in a similar way a theory for deriving left exact functors onthe left if one assumes that A has enough projectives, so that to any object Mone can functorially associate a projective resolution

. . . P 1(M)→ P 0(M)→M→ 0 .

Then for every bounded above complexM• there exists a bounded above complexP (M•) of projective objects which defines a functor P : K−(A) → K−(A). Thenthe functor LF : K−(A) → K−(B) given by LF (M•) = F (P (M•)) defines asabove a left derived functor

LF : D−(A)→ D−(B) .

Analogous properties to those proved for right derived functors hold for left derivedfunctors.


We can also derive on the right functors F : K(A) → K(B) that are notinduced by a left exact functor A → B. The theory requires the substitution ofthe injective resolution I(K•) of a complex with a resolution by complexes thatare F -acyclic in some suitable sense. Let A′ be a thick Abelian subcategory of A.We consider the homotopy categories K?

A′(A) that we have already introduced.

Definition A.58. An exact functor F : K?A′(A) → K(B) has enough acyclics if

there exists a triangulated subcategory KF(A) of K?A′(A) such that the following

conditions are fulfilled:

1. There is a functor I : K?A′(A)→ KF(A);

2. for every object M• of K?A′(A) there is a quasi-isomorphism M• → I(M•)

which is functorial on M•;

3. if J • is an object of KF(A) which is acyclic as a complex, then F(J •) is anacyclic complex of objects of B.

4

Example A.59. If A has enough injectives, the functor F : K?A′(A) → K(B) in-

duced by a right exact functor F : A→ B has enough F-acyclics. One simply takesKF(A) as the category I+(A) of bounded below complexes of injective objects ofA. 4

We shall denote by RF : K?A′(A)→ K(B) the composition of the functors I

and F , that is, RF(M•) = F(I(M•)).

Lemma A.60. RF transforms quasi-isomorphisms into quasi-isomorphisms.

Proof. Since M• → I(M•) is a quasi-isomorphism, the functor I : K?A′(A) →

KF(A) transforms quasi-isomorphisms into quasi-isomorphisms. Hence, it suf-fices to show that F : KF(A)→ K(B) transforms quasi-isomorphisms into quasi-isomorphisms as well. Let f : I• → J • be a quasi-isomorphism in KF(A). Since F

is exact, F(I•) F(f)−−−→ J • → F(Cone(f)) → F(I•)[1] is an exact triangle in K(B)and then F(Cone(f)) ' Cone(F(f)). Moreover the cone of f is an acyclic complexby Corollary A.20 and is an object of KF(A), and F(Cone(f)) ' Cone(F(f)) isacyclic so that F(f) is a quasi-isomorphism again by Corollary A.20.

By Proposition A.34, RF : K?A′(A)→ K(B) induces a functor

RF : D∗A′(A)→ D(B)

which is called right derived functor of F . We shall write as before RiF(M•) =Hi(RF(M•)), the i-th right derived functor of F applied toM•. This is an objectof the Abelian category B.

The notion of F-acyclic complex is defined as before.


Definition A.61. A complex M• is F-acyclic if the natural morphism F(M•) →RF(M•) induced by M• → I(M•) is a quasi-isomorphism. 4

One then may check that any complex in KF(A) is F-acyclic by using Corol-lary A.20. Indeed, if M• is an object of KF(A), the cone of j : M• → I(M•) isacyclic and is also in KF(A). Thus, F(Cone(j)) is mapped into an acyclic complexin K(B); moreover, F(Cone(j)) ' Cone(F(j)) because F is exact, an this impliesthat F (M•)→ RF (M•) is a quasi-isomorphism.

The right derived functor RF satisfies a derived version of de Rham’s the-orem. Let M• be a complex in D?

A′(A) and M• ' J • an isomorphism in thederived category where J • is F-acyclic. De Rham’s theorem in its derived versionis just the existence of isomorphisms in the derived category

RF(M•) ' RF(J •) ' F(J •) , (A.11)

where the second isomorphism holds because J • is F-acyclic.

Proposition A.62. The right derived functor RF : D?A′(A)→ D(B) is exact. More-

over, if

M′• →M• →M′′• →M′•[1]

is an exact triangle in D∗A′(A), we have a long exact sequence of derived functors

· · · → RiF(M′•)→ RiF(M•)→ RiF(M′′•)→Ri+1F(M′•)→ Ri+1F(M•)→ Ri+1F(M′′•)→ · · ·

Proof. One readily checks that RF commutes with the shift functor. Let us provethat it is additive. If M• and N • are complexes in K?

A′(A), the complex J • =I(M•) ⊕ I(N •) is F-acyclic and the sum morphism M• ⊕ N • → J • is a quasi-isomorphism and then an isomorphism in the derived category. By de Rham’stheorem (A.11), one has RF(M• ⊕ N •) ' RF(M•) ⊕ RF (N •). It follows thatRF(f+g) = RF(f)+RF(g) for any pair of morphisms f, g : M• → N • in D?

A′(A).

We now take an exact triangle, which we may assume of the form M• f−→ N • →Cone(f) → M•[1] where f is a morphism of complexes. By Axiom TR2 of thedefinition of a triangulated category, the commutative diagram

M•f //

N •

I(M•)

I(f) // I(N •)


can be embedded into a morphism of exact triangles

M•f //

iM•

N •

iN•

// Cone(f)

//M•[1]

iM• [1]

I(M•)

I(f) // I(N •) // Cone(I(f)) // I(M•)[1]

.

The morphisms iM• and iN• are isomorphisms in the derived category, and thenCone(f)→ Cone(I(f)) is an isomorphism in D?

A′(A) as well. Since F is exact, oneconcludes.

Assume now that F takes values in a full subcategory K∗B′(B), where B′ isa thick Abelian subcategory of B and ∗ is any of the superscripts +, −, b or none.Suppose also that C is a third Abelian category, and that G : K∗B′(B)→ K(C) isanother exact functor with enough acyclics (cf. Definition A.58).

Proposition A.63 (Grothendieck’s composite functor theorem). If F transformsF-acyclic objects into G-acyclic objects, one has:

1. G F has enough acyclics, so that its right derived functor

R(G F) : D?A′(A)→ D(C)

exists;

2. one has a natural isomorphism of derived functors R(G F) ∼→ RG RF.

Proof. For the first part, one simply takes KGF(A) = KF(A). The second partfollows from

R(G F)(M•) ' (G F)(I(M•)) = G(RF(M•)) ' RG(RF(M•)) ,

where the last isomorphism is due to de Rham’s theorem (A.11) because RF(M•)' F(I(M•)) is G-acyclic.

One can similarly develop a theory of derived functors on the left. To do so,one has to replace the second condition in Definition A.58 with the following: thereis quasi-isomorphism P (M•) → M• which is functorial in M•. The left derivedfunctor is then defined by

LF(M•) = F(P (M•))

and it has similar properties to those proved for the right derived functors.


Derived direct image

Let f : X → Y be a morphism of algebraic varieties (i.e., a morphism of schemesbetween algebraic varieties). The direct image functor f∗ : Mod(X)→Mod(Y ) isleft exact, so that it induces a right derived functor

Rf∗ : D+(Mod(X))→ D+(Mod(Y ))

described as Rf∗M• ' f∗(I•), where I• is a complex of injective OX -modulesquasi-isomorphic toM•. Under very mild conditions, the direct image of a quasi-coherent sheaf is also quasi-coherent (f has to be quasi-compact and locally offinite type); in this case, Rf∗ maps complexes with quasi-coherent cohomology tocomplexes with quasi-coherent cohomology, thus defining a functor

Rf∗ : D+qc(Mod(X))→ D+

qc(Mod(Y ))

that we denote with the same symbol. We shall assume that f satisfies theseconditions. Then, the category Qco(X) has enough injectives as well; we can de-rive f∗ : Qco(X) → Qco(Y ) and obtain a derived functor Rf∗ : D+(Qco(X)) →D+(Qco(Y )), which is naturally identified with the previous one under the equiv-alences D+(Qco(X)) ' D+

qc(Mod(X)) and D+(Qco(Y )) ' D+qc(Mod(Y )).

When f is proper, so that the higher direct images of a coherent sheaf arecoherent as well (cf. [134, Thm.3.2.1] or [141, Thm. 5.2] in the projective case),we also have a functor

Rf∗ : D+(X)→ D+(Y )

between the derived categories of complexes of quasi-coherent sheaves with coher-ent cohomology sheaves.

Finally, since the dimension of X bounds the number of higher direct imagesof a sheaf of OX -modules, Rf∗ maps complexes with bounded cohomology tocomplexes with bounded cohomology, thus defining a functor

Rf∗ : Dbqc(Mod(X))→ Db

qc(Mod(Y )) .

We check now that Rf∗ can be extended to a functor

Rf∗ : D(Mod(X))→ D(Mod(Y ))

between the whole derived categories. If M• is a complex of OX -modules, wedenote by C•(M•) the double complex obtained by associating to each term inM• its canonical Godement resolution (cf. [124]). Moreover, we write C(M•) forthe associated simple complex. Since for any x ∈ X the cone of M•

x → C(M•)xis homotopic to zero, the natural map M• → C(M•) is a quasi-isomorphism.Furthermore, C(M•) is a complex of flabby sheaves, since the infinite direct sum


of flabby sheaves is flabby on Noetherian spaces. Finally, if K• is an acyclic complexof flabby sheaves, the complex f∗K• is acyclic. This follows easily from the longexact sequence of higher direct images and the fact that the dimension of X boundsthe number of higher direct images of a sheaf of OX -modules. In conclusion, wecan take the category of complexes of flabby OX -modules as a category of f∗-acyclic complexes (Definition A.58), and define the right derived functor Rf∗by Rf∗(M•) = f∗(C(M•)). This functor maps Dqc(Mod(X)) to Dqc(Mod(Y ))and also Db

qc(Mod(X)) to Dbqc(Mod(Y )). If f is proper, it maps Db

c(Mod(X)) toDbc(Mod(Y )) and Db(X) to Db(Y ).

Proposition A.64. Let f : X → Y be a morphism of algebraic varieties and M• ∈D(Mod(X)) a complex of OX-modules. Then

Rf∗M• ' lim−→n

Rf∗(M•≤n)

Proof. By construction, C(M•) ' lim−→nC(M•

≤n). One concludes as on Noethe-rian spaces f∗ commutes with direct limits [124].

If g : Y → Z is another morphism of algebraic varieties, then (gf)∗ = g∗f∗and one may apply Grothendieck’s composite functor theorem (since f∗ transformsflabby sheaves into flabby sheaves), obtaining

R(g f)∗ ' Rg∗ Rf∗ .

If Y is a point, Mod(Y ) is the category of k-vector spaces and f∗ is the functorof global sections Γ(X, ). In this case, Rf∗M• = RΓ(X,M•) and Rif∗M• iscalled the i-th hypercohomology group Hi(X,M•) of the complexM•. It coincideswith the cohomology group Hi(X,M) when the complex reduces to a single sheaf.

Derived homomorphism functor

Let A be an Abelian category with enough injectives (in the strong sense requiredin Section A.4.5). We wish to construct a “derived functor” of the complex ofhomomorphisms L• 7→ F(L•) = Hom•(K•,L•) for a fixed complex K•. Since thisfunctor is not induced by a left-exact functor A → B, we have to find a suitablecategory of F-acyclics.

Definition A.65. A complex I• is injective if the functor Hom•( , I•) takes quasi-isomorphisms into quasi-isomorphisms. 4

Since the functor Hom•( , I•) transforms the cone of a morphism R• →M•

into the cone of Hom•(M•, I•)→ Hom•(R•, I•), a complex I• is injective if andonly if it transforms acyclic complexes into acyclic complexes.


Lemma A.66.

1. Any bounded below complex of injective OX-modules is injective.

2. If I• is injective and acyclic, then Hom•(M•, I•) is acyclic for any complexM•.

Proof. 1. Let J • be a bounded below complex of injective OX -modules andM• anacyclic complex. One has to prove that any morphism of complexes f : M• → J •is homotopic to zero. We construct a homotopy operator proceeding by recur-rence. Assume that for every i ≤ r we have constructed a morphism hi : Mi →J i−1 satisfying di−1 hi + hi+1 di = f i for any i < r. The morphism g =fr − dr−1 hr : Mr → J r vanishes on im dr−1 = ker dr, and induces a mor-phismMr/ ker dr → J r. Since J r is injective, this morphism lifts to a morphismhr+1 : Mr+1 → J r, and one has dr−1 hr + hr+1 dr = fr.

2. The complex Hom•(I•, I•) is acyclic, and in particular, the identity mor-phism is homotopic to zero, i.e., I• is homotopic to zero. Hence Hom•(M•, I•) isalso homotopic to zero.

Lemma A.67. Let I• be an injective complex. For any complex M• the naturalmorphism

HomK(A)(M•, I•)→ HomD(A)(M•, I•)

is an isomorphism.

Proof. It is enough to see that if R• →M• is a quasi-isomorphism, any morphismof complexes R• → I• lifts (up to homotopies) to a morphism of complexesM• →I•. Since Hom•(M•, I•) → Hom•(R•, I•) is a quasi-isomorphism, taking H0 oneconcludes.

Let I+(A) be the full subcategory of K+(A) formed by the bounded abovecomplexes of injective objects. Recall that there is a functor I : K+(A) → I+(A)and a natural quasi-isomorphism M• → I(M•), which depends functorially onM•. It follows from Lemma A.66 that for any complex K•, the functor F(L•) =Hom•(K•,L•) has enough injectives and one can take for KF(A) the categoryI+(A) of bounded above complexes of injective objects of A. Therefore, thereexists a right derived functor

RIIHom•(M•, ) : D+(A)→ D(Ab) .

(The subscript “II” reflects the fact that we are deriving with respect to the secondvariable.) By using Lemma A.66, one proves that for any fixed object L• ∈ D+(X),the functor

RIIHom•( ,N •) : K(A)0 → D(Ab)


maps quasi-isomorphisms to isomorphisms, so that it induces, again by Proposi-tion A.34, a functor RIRIIHom•( ,N •) : D(A)0 → D(Ab). Hence one obtains abifunctor

RHom•X : D(A)0 ×D+(A)→ D(Ab)

defined as RHom•(M•,N •) = RIRIIHom•(M•,N •) ' Hom•(M•, I•)), whereI• is complex of injective sheaves quasi-isomorphic to N •. We shall denote

Exti(M•,N •) = RiHom•(M•,N •) = Hi(RHom•(M•,N •)) .

If A has enough projectives, one can derive the homomorphisms in the reverseorder than before, so that for any complex N • we have a right derived functor

RIHom•( ,N •) : D−(A)0 → D(Ab)

given by RIHom•(M•,N •) ' Hom•(P (M•),N •), where P (M•)→M• is a pro-jective resolution. Moreover, this functor induces a bifunctor

RIIRIHom• : D−(A)0 ×D(A)→ D(Ab) .

If A has both enough injectives and projectives, the functors RIRIIHom• andRIIRIHom• coincide over D−(A)0 ×D+(A).

The following property, known as Yoneda’s formula, holds true.

Proposition A.68. Let M• be a complex and N • a bounded below complex. Onehas

Exti(M•,N •) ' HomiD(A)(M•,N •) ,

where we have written HomiD(A)(M•,N •) = HomD(A)(M•,N •[i]).

Proof. One has

Exti(M•,N •) ' Hi(RHom•(M•,N •)) ' Hi(Hom•(M•, I(N •)))' HomK(A)(M•, I(N •)[i]) ' HomD(A)(M•, I(N •[i]))' HomD(A)(M•,N •[i])

where the fourth isomorphism is due to Lemma A.67.

We shall also use the notation

HomiB(a, b) = HomB(a, b[i]) . (A.12)

for objects a and b of any triangulated category B.

Eventually, we consider the case when A is one of the categories Mod(X) orQco(X) associated with an algebraic variety X over an algebraically closed field


k. In these cases we write HomX(M,N ) for HomA(M,N ) and the same for thecorresponding complexes of homomorphisms. We have a bifunctor

RHom•X : D(Mod(X))0 ×D+(Mod(X))→ D(Ab) ,

which now takes values in the derived category D(k) of k-vector spaces.

One can also consider the complex of sheaves of homomorphisms, which wedenote by Hom•OX (M•,N •). This is given by

Homn(M•,N •) =∏i

HomOX (Mi,N i+n)

with the differential df = f dM• + (−1)n+1dN• f . Proceeding as above we candefine a derived sheaf homomorphism

RHom•OX = RIRIIHom•OX : D(Mod(X))0 ×D+(Mod(X))→ D(Mod(X))

described as

RHom•OX (K•,L•) ' RIRIIHom•OX (K•,L•)' Hom•OX (K•, I•) ,

where I• is a bounded below complex of injective objects quasi-isomorphic to L•.The total derived bifunctor RHom•OX induces bifunctors

RHom•OX : D(X)0 ×D+(Qco(X))→ D(Qco(X))

RHom•OX : D−(X)0 ×D+(X)→ D(X) .

We can apply Grothendieck’s composite functor theorem to the compositionΓ(U,Hom•OX (K•,L•)) ' Hom•OU (K•|U ,L•|U ), to obtain an isomorphism in thederived category D(Mod(U)):

RΓ(U,RHom•OX (K•,L•)) ' RHom•OU (K•|U ,L•|U ) .

The categories Mod(X), Qco(X) and Coh(X) do not have enough projectives.However, in some situations one can derive the local homomorphisms first withrespect to the first argument and then with respect to the second. One such caseoccurs when X has the resolution property, that is, every coherent sheaf admitsa resolution by locally free sheaves (possibly of infinite rank). This happens, forinstance, in the following cases:

• X is smooth;

• X is quasi-projective.


The problem is now circumvented by considering complexes P• of locally freesheaves. For every bounded above complex M• of OX -modules with coherentcohomology there exist a bounded above complex P (M•) of locally free sheavesof finite rank and a quasi-isomorphism P (M•)→M•. Then, for any complex N •we have a right derived functor RIHom•OX ( ,N •) : D−c (Mod(X))0 → D(Mod(X))given by RIHom•OX (M•,N •) ' Hom•OX (P (M•),N •). Moreover, this induces abifunctor

RIIRIHom•OX : D−c (Mod(X))0 ×D(Mod(X))→ D(Mod(X)) .

The derived functors RIRIIHom•OX and RIIRIHom•OX coincide on the productD−c (Mod(X))0 ×D(Mod(X)).

Yoneda product

We describe here the Yoneda product between Hom groups in a triangulated cat-egory B. Let a, b, c be objects in B. The composition of Hom groups gives mor-phisms

HomB(a, b[i])×HomB(b[i], c[k])→ HomB(a, c[k]) .

The shift functor yields an isomorphism HomB(b[i], c[k]) ' HomB(b, c[k − i]);therefore we get the Yoneda product

Yij : HomB(a, b[i])×HomB(b, c[j])→ HomB(a, c[i+ j])

or, equivalently,

Yij : HomiB(a, b)×Homj

B(b, c)→ Homi+jB (a, c) . (A.13)

When B is a k-linear category, this product is actually bilinear.

Whenever B is the derived category of an Abelian category A, by Yoneda’sformula (see Proposition A.68) we may write this product in the more usual formas

Exti(M•,N •)× Extj(N •,Q•)→ Exti+j(M•,Q•)

where M•, N •, Q• are objects in the derived category D(A), with N • and Q•bounded below.

One should notice that any exact functor of triangulated categories F : B→C, being compatible with compositions and shifts, yields a commutative diagram

HomiB(a, b)×Homj

B(b, c)Yij //

F×F

Homi+jB (a, c)

F

Homi

C(F (a), F (b))×HomjC(F (b), F (c))

Yij // Homi+jC (F (a), F (c)) .

(A.14)


Extensions in triangulated categories

If M and N are objects of an Abelian category A, the elements of the groupExt1(M,N ) can be identified with the extensions of M by N , that is, with theexact sequences

0→ N → F →M→ 0 ,

where two extensions 0 → N → F → M → 0 and 0 → N → F ′ → M → 0are equivalent if there is an isomorphism φ : F → F ′ fitting into a commutativediagram

0 // N // F //

φ

M // 0

0 // N // F ′ //M // 0

.

Taking Proposition A.68 into account, one has Ext1(M,N ) ' Hom1D(A)(M,N )

so that Hom1D(A)(M,N ) is identified with the group of extensions ofM by N . We

are going to check that this identification still holds true if we take an arbitrarytriangulated category B instead of D(A). Let a, b be objects of B.

Definition A.69. An extension of b by a is an exact triangle

a→ c→ bf−→ a[1] .

Two extensions a→ c→ bf−→ a[1], a→ c′ → b

f ′−→ a[1] are said to be equivalent ifthere is an isomorphism of triangles

a // c //

φ'

bf // a[1]

a // c′ // bf ′ // a[1] .

4

So f ′ = f , and an equivalence class of extensions gives rise to a uniquemorphism f ∈ HomB(b, a[1]) = Hom1

B(b, a), cf. (A.12). Conversely, by axiomsTR1 and TR2 any morphism f : b→ a[1] can be embedded in an exact triangle

a→ c→ bf−→ a[1] ,

and two such triangles give rise to isomorphic extensions due to axiom TR4. Sowe have the following result.

Proposition A.70. Given two objects a and b of a triangulated category B, the setof equivalence classes of extensions of b by a can be naturally identified with thegroup Hom1

B(b, a).


Derived tensor product and inverse image

Let X be an algebraic variety. We want to derive on the left the functor “tensorproduct of complexes.” The lack of projectives in Mod(X) is here circumventedby considering flat sheaves. We say that a complex P• of OX -modules is flat iffor any acyclic complex N •, the tensor product complex N • ⊗ P• is also acyclic.This amounts to saying that the functor ⊗P• transforms quasi-isomorphisms toquasi-isomorphisms. One can readily check that a bounded complex P• is flat ifand only if every sheaf Pn is a flat OX -module. For unbounded complexes, wehave the following result.

Lemma A.71. If P• is a bounded above complex of flat OX-modules, then P• isflat.

Proof. Let N • be an acyclic complex. One has N • ' lim−→nN •≤n and an isomor-

phism of bicomplexes N • ⊗ P• ' lim−→n(N •≤n ⊗ P•) (cf. Example A.23). Since

cohomology commutes with direct limits, we may assume that N • is boundedabove. Now, Hd1(N • ⊗ P•) = Hd1(N •) ⊗ P• = 0 and by Proposition A.27, the(simple) complex N • ⊗ P• is acyclic.

Lemma A.72. Any OX-moduleM is a quotient of a flat OX-module P (M), whichdepends functorially on M.

Proof. For each open subset U , let OX,U be the OX -module defined by lettingOX,U (V ) be the subgroup of sections s ∈ OX(V ) with support contained in U ∩Vfor any open subset V ⊆ X. The sheaf OX,U is flat for every open subset U ,because the stalk of OX,U at a point x is the local ring OXx if x ∈ U and zerootherwise. Moreover HomX(OX,U ,M) = Γ(U,M). Then, taking a copy of OX,Ufor each nonzero section in Γ(U,M) and the direct sum P (M) = ⊕OX,U over allthe open sets U and all such sections, we obtain an epimorphism P (M)→M→ 0.We also set P (0) = 0. The functoriality follows from the construction.

It follows thatM admits a resolution P•(M)→M→ 0 by flat OX -modules.By Lemma A.71, the complex P•(M) is flat. For any bounded above complex N •,let P (N •) be the simple complex associated with the double complex P•(N q),and for any complex N • let us define P (N •) = lim−→n

P (N •≤n). We then have afunctor

P : C(A)→ C(A) .

Lemma A.73. The natural morphism P (N •) → N • is a quasi-isomorphism, andthe complex P (N •) is flat.

Proof. Assume at first that N • is bounded above. By Lemma A.71, P (N •) isflat, and by Proposition A.27, P (N •) → N • is a quasi-isomorphism as claimed.


In the general situation, the morphism P (N •) → N • is a quasi-isomorphism,since cohomology commutes with direct limits. Finally, P (N •) is flat because bothtensor product and cohomology commute with direct limits.

Corollary A.74. Let P• be a flat and acyclic complex. Then N •⊗P• is acyclic forany complex N •.

Proof. Since P (N •)→ N • is a quasi-isomorphism and P• is flat, P (N •)⊗ P• →N •⊗P• is a quasi-isomorphism as well. One concludes because P (N •) is flat andP• is acyclic.

It follows that for a fixed M• ∈ K(Mod(X)) the functor “tensor product ofcomplexes”

M•⊗ : K(Mod(X))→ K(Mod(X))

has enough acyclics (one has to take for KM•⊗ the category of flat complexes).

Thus, there exists a left derived functor, denoted by M•⊗ : D(Mod(X)) →D(Mod(X)). Now, for a fixedN •, the functor ⊗N • : K(Mod(X))→ D(Mod(X)),induces a bifunctor, called derived tensor product

L⊗ : D(Mod(X))×D(Mod(X))→ D(Mod(X))

whose description is M•L⊗N • ' M• ⊗ P (N •), where P (N •) → N • is a quasi-

isomophism and P (N •) is a complex of flat sheaves.

One can derive the tensor product reversing the sense of the derivations and

obtaining the same result, i.e.,M•L⊗N • ' N •

L⊗M•. It is also easy to check that

(M•L⊗N •)

L⊗P• 'M•

L⊗ (N •

L⊗P•) .

The derived tensor product induces functors

Dqc(Mod(X))×Dqc(Mod(X))L⊗−→ Dqc(Mod(X))

D−c (Mod(X))×D−c (Mod(X))L⊗−→ D−c (Mod(X)) .

In order to get a derived tensor product for the category D(Qco(X)), one hasto modify the previous approach, as for a complex M• of quasi-coherent sheavesthe flat resolution P (M•) may fail to be a complex of quasi-coherent sheaves.Nonetheless it is possibly to construct a different flat resolution Q(M•) whichis a complex of quasi-coherent sheaves. In the affine case, one sets Q(M•) =lim−→n

Q(M•≤n), where Q(M•

≤n) → M•≤n is a resolution by free (possibly of


infinite rank) sheaves. The general case can be handled by using Cech resolutionsassociated with an affine covering. In this way the functor

D(Qco(X))×D(Qco(X))L⊗−→ D(Qco(X))

is defined. Moreover, it induces a functor

D−(X)×D−(X)L⊗−→ D−(X) .

Proposition A.75. Let K•, L•, M• be complexes of OX-modules with M• boundedbelow. One has an isomorphism of k-vector spaces

HomD(Mod(X))(K•L⊗L•,M•) ' HomD(Mod(X))(K•,RHom•OX (L•,M•)) .

Moreover, if L• is bounded above, one has an isomorphism in the derived categoryD(Mod(X))

RHom•OX (K•L⊗L•,M•) ' RHom•OX (K•,RHom•OX (L•,M•)) .

Proof. One has an isomorphism between the homomorphism complexes

Hom•(K• ⊗ L•,M•) ' Hom•(K•,Hom•OX (L•,M•)) .

In particular, if L• is flat and M• is injective, Hom•(L•,M•) is injective. Now,let P• → L• be a flat resolution andM• → I• an injective resolution. By LemmaA.67 and Equation A.3,

HomD(Mod(X))(K•L⊗L•,M•) ' H0(Hom•(K• ⊗ P•, I•)

' H0(Hom•(K•,Hom•OX (P•, I•))) .

Since P• is flat and I• is injective, Hom•OX (P•, I•) is an injective resolution ofRHom•OX (L•,M•). By applying again Lemma A.67, one concludes. For the secondpart, it is enough to prove that for any complex R• of OX -modules, one has

HomD(Mod(X))(R•,RHom•OX (K•L⊗L•,M•))

' HomD(Mod(X))(R•,RHom•OX (K•,RHom•OX (L•,M•)))

which follows easily from the first statement.

Definition A.76. A complex of OX -modulesM• is of finite Tor-dimension if thereexist a bounded and flat complex P (M•) and a quasi-isomorphism P (M•)→M•.

4


Complexes of finite Tor-dimension are characterized by the following result,whose proof is straightforward.

Proposition A.77. A complex of OX-modulesM• is of finite Tor-dimension if andonly if the functor M•⊗ maps Db(Mod(X)) to Db(Mod(X)).

Moreover for complexes with coherent cohomology, finite Tor-dimension isequivalent to finite homological dimension.

Proposition A.78. Let M• be an object in Db(X). The following conditions areequivalent:

1. M• is of finite homological dimension.

2. M• is of finite Tor-dimension.

3. RHom•OX (M•,G•) is in Db(X) for every G• in Db(X).

Proof. The three conditions are local so that we can assume that X is affine. It isclear that (1) implies (2) (by Proposition A.77) and (3). We check that (3) implies(1). Let us consider a quasi-isomorphism L• →M•, where L• is a bounded abovecomplex of finitely generated free modules. If Kn is the kernel of the differentialLn → Ln+1, for n small enough the truncated complex Kn → Ln → . . . isquasi-isomorphic to M• becauseM• is an object of Db(X). Let x be a point andOx its residual field. Since RHom•OX (M•,Ox) has bounded homology, one hasExt1

OX (Kn,Ox) = 0 for n small enough. For such an n the module Kn is free in aneighborhood of x and one concludes. To prove that (2) implies (1), one proceedsanalogously by replacing Ext1 with Tor1.

Let f : X → Y be a morphism of algebraic varieties. The inverse image leftexact functor f∗ : Mod(Y )→Mod(X) can be derived on the left in a similar wayto the tensor product of complexes. We take Kf∗(Mod(Y )) as the category of flatcomplexes of OY -modules, and obtain a left derived functor

Lf∗ : D(Mod(Y ))→ D(Mod(X))

given by Lf∗(M•) = f∗(P (M•)), provided that P (M•) is a flat complex of OY -modules and P (M•) → M• a quasi-isomorphism. One readily checks that Lf∗

induces a functor Lf∗ : Dqc(Mod(Y )) → Dqc(Mod(X)). To define the functorLf∗ : D(Qco(Y )) → D(Qco(X)), one has to proceed as in the case of the de-rived tensor product for complexes of quasi-coherent sheaves. In this way we geta functor Lf∗ : D(Y ) → D(X). All these morphisms map bounded above com-plexes to bounded above complexes. When f is a flat morphism, there is a naturalisomorphism of functors f∗ ' Lf∗.


If g : Z → X is another morphism, g∗ transforms flat complexes to flat com-plexes. By Proposition A.63, one has an isomorphism of functors

L(f g)∗ ' Lg∗ Lf∗ .

Compatibility between the derived tensor product and the derived inverseimage is easily checked.

Proposition A.79. Let f : X → Y be a morphism of algebraic varieties. If M• andN • are complexes in D(Mod(Y )), there is a functorial isomorphism

(Lf∗M•)L⊗ (Lf∗N •) ∼→ Lf∗(M•

L⊗N •) .

Proof. P (M•)⊗ P (N •)→M• ⊗N • is a flat resolution. Since f∗ takes flat com-plexes to flat complexes, the natural isomorphism

f∗(P (M•)⊗ P (N •)) ∼→ f∗P (M•)⊗ f∗P (N •)

induces an isomorphism Lf∗(M•L⊗N •) ∼→ (Lf∗M•)

L⊗ (Lf∗N •).

In some situations the derived inverse image induces a functor

Lf∗ : Db(Y )→ Db(X) .

This is the case, for instance, when every coherent sheaf G on Y admits a finiteresolution by coherent locally free sheaves, a condition which, by Serre’s criterion,is equivalent to the smoothness of Y . In this hypothesis, every objectM• in Db(Y )can be represented as a bounded complex L• of coherent locally free sheaves sothat Lf∗M• ' f∗L• is bounded. Another example occurs when f has finite Tor-dimension, that is, when for every coherent sheaf G on Y there are only a finitenumber of nonzero derived inverse images Ljf∗(G) = H−j(Lf∗(G)); in particular,flat morphisms have finite Tor-dimension.

A.4.6 Some remarkable formulas in derived categories

In this section we gather together some formulas in the derived category which areused throughout this book. We recall that if f : X → Y is a morphism of algebraicvarieties, the dimension of X bounds the number of higher direct images of a sheafof OX -modules; in this case we say that f has finite cohomological dimension. Itfollows that Rf∗ maps complexes with bounded cohomology to complexes withbounded cohomology.

The first result gives the adjunction formula between the derived inverse andthe derived direct images.


Proposition A.80. Let f : X → Y be a morphism of algebraic varieties. One hasan isomorphism of k-vector spaces

HomD(Mod(X))(Lf∗M•,N •) ' HomD(Mod(Y ))(M•,Rf∗N •) ,

where M• is a complex of OY -modules and N • is a complex of OX-modules.

Proof. Every quasi-isomorphism N • → R• of complexes of OX -modules is domi-nated by an f∗-acyclic complex. To see this it is enough to consider an f∗-acyclicresolution of R•. As a consequence of Equation (A.7), Lemma A.67 and Equation(A.3), it follows that

HomD(Mod(X))(L•,N •) ' lim−→I•H0(Hom•(L•, I•)) ,

where I• runs over the f∗-acyclic resolutions of N •. Let P• → M• be a flatresolution. In particular we have,

HomD(Mod(X))(Lf∗M•,N •) ' lim−→I•H0(Hom•(f∗P•, I•)) .

Since this isomorphism holds for any flat resolution of M•, one has

HomD(Mod(X))(Lf∗M•,N •) ' lim−→I•,P•

H0Hom•(f∗P•, I•) .

The usual adjunction formula for sheaves yields Hom•(f∗P•, I•) ' Hom•(P•, f∗I•).By remarking that every resolution R• →M• is dominated by a flat one, for ex-ample by P (R•), one can similarly conclude that

lim−→I•,P•

H0Hom•(P•, f∗I•) ' HomD(Mod(Y ))(M•,Rf∗N •) .

Corollary A.81. Let f : X → Y be a morphism of algebraic varieties and N • abounded below complex of OY -modules. One has a functorial isomorphism

τ : Rf∗RHom•OX (Lf∗M•,N •) ' RHom•OY (M•,Rf∗N •) .

Proof. For any K• ∈ D(Mod(Y )), one has isomorphisms of k-vector spaces

HomD(Mod(Y ))(K•,Rf∗RHom•OX (Lf∗M•,N •)' HomD(Mod(X))(Lf∗K•,RHom•OX (Lf∗M•,N •)

' HomD(Mod(X))(Lf∗K•L⊗Lf∗M•,N •)

' HomD(Mod(X))(Lf∗(K•L⊗M•),N •)

' HomD(Mod(Y )))(K•L⊗M•,Rf∗N •)

' HomD(Mod(Y ))(K•,RHom•OY (M•,Rf∗N •)) .


One concludes by Yoneda’s Lemma A.2.

Corollary A.82. If f has finite Tor-dimension, the functor Lf∗ : Db(Y )→ Db(X)is left adjoint to Rf∗ : Db(X)→ Db(Y ).

Proposition A.83 (Projection formula). Let f : X → Y be a morphism of algebraicvarieties. For every complex M• of OX-modules and every complex N • of OY -modules, there is a functorial morphism in D(Mod(Y ))

Rf∗(M•)L⊗N • → Rf∗(M•

L⊗Lf∗N •) ,

which is an isomorphism if N • has quasi-coherent cohomology.

Proof. Let C(M•) ⊗ f∗P (N •) → J • be a quasi-isomorphism, where J • is a f∗-acyclic complex. One has morphisms

f∗(C(M•))⊗ P (N •)→ f∗(C(M•)⊗ f∗P (N •))→ f∗J • ,

where the first morphism is the projection formula for sheaves. Hence one has amorphism

Rf∗(M•)L⊗N • → Rf∗(M•

L⊗Lf∗N •)

in the derived category. We prove that this is an isomorphism if N • has quasi-coherent cohomology. Since the question is local on Y and N • ' lim−→n

N •≤n,we may assume that Y is affine and that N • is bounded above. There is a quasi-isomorphism L• → N •, where L• is a bounded above complex of free OY -modules.

Then C(M•)⊗ f∗L• is a f∗-acyclic resolution of M•L⊗Lf∗N •; we need to check

that f∗(C(M•))⊗L• → f∗(C(M•)⊗ f∗L•) is a quasi-isomorphism, which followsstraightforwardly from the fact that L• is a complex of free OY -modules (actually,this is an isomorphism).

Proposition A.84. Let f : X → Y be a flat morphism of algebraic varieties. IfM• ∈ D−c (Mod(Y )) and N • ∈ D+

c (Mod(Y )), one has a functorial isomorphism

f∗RHom•OY (M•,N •) ∼→ RHom•OX (f∗M•, f∗N •)

in the derived category.

Proof. Let N • → I• be an injective resolution. One has natural morphisms

f∗Hom•(M•, I(N •))→ Hom•(f∗M•, f∗I(N •))→ Hom•(f∗M•,J •) ,

where f∗I(N •) → J • is a quasi-isomorphism and J • is a complex of injectiveOX -modules. Hence we have a morphism

f∗RHom•(M•,N •)→ RHom•(f∗M•, f∗N •) .


To prove that this is an isomorphism, we may assume that Y is affine. There isa quasi-isomorphism L• → M•, where L• is a bounded above complex of freemodules of finite rank, and we have only to check that

f∗Hom•(L•,N •)→ Hom•(f∗L•, f∗N •)

is a quasi-isomorphism. This follows from a direct computation (which indeedproves that this is an isomorphism).

One of the most useful formulas in connection with the Fourier-Mukai trans-form is the base change formula in derived category.

Proposition A.85. Let us consider a Cartesian diagram of morphisms of algebraicvarieties

X ×Y Yg //

f

X

f

Y

g // Y

.

For any complex M• of OX-modules there is a natural morphism

Lg∗Rf∗M• → Rf∗Lg∗M• .

Moreover, ifM• has quasi-coherent cohomology and either f or g is flat, the abovemorphism is an isomorphism.

Proof. The natural morphism M• → Rg∗Lg∗M• induces a morphism

Rf∗M• → Rf∗Rg∗Lg∗M• ' Rg∗Rf∗Lg∗M•

and, by adjunction, a morphism in the derived category

Lg∗Rf∗M• → Rf∗Lg∗M• .

We can prove that this is an isomorphism if M• has quasi-coherent cohomology.The question is local, so that we can assume that both Y and Y are affine, and theng and g are affine morphisms. It is enough to check that the induced morphism

Rg∗(Lg∗Rf∗M•)→ Rg∗(Rf∗Lg∗M•) ' Rf∗(Rg∗Lg∗M•)

is an isomorphism. By the projection formula (Proposition A.83), this is equivalentto proving that the morphism

Rf∗M•L⊗ g∗OeY → Rf∗(M•

L⊗ g∗O eX)


is an isomorphism. Again by the projection formula, the latter morphism is anisomorphism whenever

g∗(O eX) ' Lf∗(g∗OeY ) .

This happens if either f or g is flat, since in both cases the last equation reducesto the isomorphism g∗(O eX) ' f∗(g∗OeY ).

Proposition A.86. Let X be an algebraic variety, M• a bounded above complexof OX-modules with coherent cohomology, and N • and H• bounded below com-plexes of OX-modules. Assume either that H• has finite homological dimension orthat M• has finite homological dimension and X has the resolution property (forinstance, X smooth or quasi-projective). Then one has a functorial isomorphism

RHom•OX (M•,N •)L⊗H• ' RHom•OX (M•,N •

L⊗H•)


Proof. Note that in both cases, the two sides of the isomorphism are well defined.We prove first that there exists a morphism

RHom•OX (M•,N •)L⊗H• → RHom•OX (M•,N •

L⊗H•)

in the derived category. Let us consider the case when H• has finite homologicaldimension. Let P• → H• be a quasi-isomorphism where P• is a bounded complexof flat sheaves. If I(N •)⊗P• → J • is a quasi-isomorphism, where J • is a boundedbelow complex of injective sheaves, one has natural morphisms

Hom•OX (M•, I(N •))⊗ P• → Hom•(M•, I(N •)⊗ P•)→ Hom•OX (M•,J •)

and hence a morphism in the derived category

RHom•OX (M•,N •)L⊗H• → RHom•OX (M•,N •

L⊗H•) .

The proof of the existence of such a morphism when M• has finite homologicaldimension and X has the resolution property is done in a similar way. To checkthat this is an isomorphism we can proceed locally, so that we can assume thatX is affine. Then there is a quasi-isomorphism L• →M•, where L• is a boundedabove complex of free modules of finite rank, and we can use L• to derive thehomomorphisms on the first variable. We are thus reduced to check that there isan isomorphism of complexes

Hom•OX (L•,N •)⊗ P• → Hom•OX (L•,N • ⊗ P•) ,

which is a direct computation.


In the remainder of this section we assume thatX has the resolution property,i.e., any coherent sheaf on X is a quotient of a locally free sheaf of finite rank (forinstance, X smooth or quasi-projective).

Proposition A.87. Let K• be a complex of finite homological dimension. The de-rived dual

K•∨ = RHom•OX (K•,OX)

has finite homological dimension and one has functorial isomorphisms K• ' K•∨∨

and K•∨L⊗N • ' RHom•OX (K•,N •) in the derived category, where N • is a bounded

below complex of OX-modules. Moreover

RHom•OX (M•,N •)L⊗K• ' RHom•OX (M•,N •

L⊗K•)

' RHom•OX (M•L⊗K•∨,N •)

for every bounded above complex M• of OX-modules.

Proof. By hypothesis, every point x has an open neighborhood U such that thereexists a quasi-isomorphism L• → K•|U , where L• is a bounded complex of finitelocally free OU -modules. Then K•∨|U ' Hom•(L•,OU ), and the natural morphismK•|U → K•∨∨|U is an isomorphism. This proves that K•∨ has finite homological

dimension and that K• ' K•∨∨. The isomorphism K•∨L⊗N • ' RHom•OX (K•,N •)

follows from Proposition A.86. Finally, one has

RHom•OX (M•,N •)L⊗K• ' RHom•OX (K•∨,RHom•OX (M•,N •))

' RHom•OX (K•∨L⊗M•,N •))

' RHom•OX (M•,RHom•OX (K•∨,N •))

' RHom•OX (M•,N •L⊗K•) .

A straightforward consequence is the following result.

Corollary A.88. Let K• be a complex of finite homological dimension. The functorL⊗K•∨ : Db(X)→ Db(X) is both left and right adjoint to the functor

L⊗K• : Db(X)

→ Db(X), that is, there are functorial isomorphisms

HomDb(X)(M•,N •L⊗K•) ' HomDb(X)(M•

L⊗K•∨,N •)

HomDb(X)(M•,N •L⊗K•∨) ' HomDb(X)(M•

L⊗K•,N •)

where M• and N • are bounded complexes of quasi-coherent sheaves with coherentcohomology.


We now prove a Kunneth formula in the derived category. Let X and Y

be algebraic varieties. We denote by πX and πY the projections of X × Y ontoits factors. Given complexes M• of OX -modules and N • of OY -modules, we canconsider their box product

M•LN • = π∗XM•

L⊗π∗YN • .

in the derived category D(Mod(X × Y )).

Theorem A.89. If X and Y are smooth and projective, one has an isomorphism

HomhDb(X×Y )(M

•LN •, E•

LF•) '⊕

i+j=h

HomiDb(X)(M

•, E•)⊗HomjDb(Y )

(N •,F•)


Proof. Let us denote by p : X → Spec k and q : Y → Spec k the projections ontoa point, so that we have a base change diagram

X × YπY //

πX

Y

q

X

p // Spec k

.

One has

RHom•X×Y (M•LN •, E•

LF•) ' Rp∗RπX∗RHom•OX×Y (M•

LN •, E•

LF•) .

We now compute the right-hand side of this formula.

RπX∗RHom•OX×Y (π∗XM•L⊗π∗YN •, π∗XE•

L⊗π∗Y F•)

' RπX∗RHom•OX×Y (π∗XM•,RHom•OX×Y (π∗YN •, π∗XE•L⊗π∗Y F•))

' RπX∗RHom•OX×Y (π∗XM•,RHom•OX×Y (π∗YN •, π∗Y F•)L⊗π∗XE•)

' RπX∗RHom•OX×Y (π∗XM•, π∗YRHom•OY (N •,F•)L⊗π∗XE•)

' RHom•OX (M•,RπX∗[π∗YRHom•OY (N •,F•)L⊗π∗XE•])

' RπX∗RHom•OX (M•,RπX∗[π∗YRHom•OY (N •,F•)]L⊗E•)


by Propositions A.75, A.86, A.80, and the projection formula (Proposition A.83).We may use these results in view of the smoothness hypothesis. Moreover one has

RHom•OX (M•,RπX∗[π∗YRHom•OY (N •,F•)]L⊗E•)

' RHom•OX (M•, E•)L⊗ p∗[Rq∗RHom•OY (N •,F•)]

by Proposition A.86, and then

RHom•Db(X×Y )(M•

LN •, E•

LF•)

' Rp∗(RHom•OX (M•, E•)L⊗ p∗[Rq∗RHom•OY (N •,F•)])

' Rp∗RHom•OX (M•, E•)L⊗Rq∗RHom•OY (N •,F•)

' RHom•Db(X)(M•, E•)

L⊗RHom•Db(Y )(N

•,F•)) ,

again by the projection formula. We finish by applying Yoneda’s formula (Propo-sition A.68).

A.4.7 Support and homological dimension

In this section we recall results about the support and the homological dimensionof an object of the bounded derived category taken from [68] and [71].

Let X be an algebraic variety which has the resolution property.

Definition A.90. The support Supp(F•) of a complex F• in Db(X) is the unionof the supports of all its cohomology sheaves Hi(F•). 4

The support is easily characterized in terms of morphisms to the structuresheaves Ox of the (closed) points.

Proposition A.91. A (closed) point x ∈ X is in the support of an object F• ofDb(X) if and only if Homi

D(X)(F•,Ox) 6= 0 for some integer i.

Proof. There is a spectral sequence Ep,q2 = ExtpX(H−q(F•),Ox) converging toEp+q∞ = Homp+q

D(X)(F•,Ox). If x ∈ Supp(F•), let q0 be the maximum of the q’s

such that x belongs to the support of H−q(F•). Then there is a nonzero morphismH−q0(F•) → Ox which gives an element of E0,q0

2 that survives to infinity; thusHom−q0D(X)(F

•,Ox) 6= 0. The converse is evident.

Remark A.92. An analogous argument shows that if there is an integer i0 suchthat Homi

D(X)(F•,Ox) = 0 for i < i0, then Hi(F•) = 0 at the point x for i < i0.

4


The length of a bounded complex 0→ En → · · · → En+m → 0 (n ∈ Z) is thenumber m.

Since every coherent sheaf on X is a quotient of a locally free sheaf of finiterank, every complex F• in Db(X) is quasi-isomorphic to a bounded above complexE• of locally free sheaves of finite rank.

Definition A.93. If F• is of finite homological dimension (Definition A.42), theminimum hd(F•) of the lengths m of the complexes of locally free sheaves iso-morphic with it in the derived category is called the homological dimension ofF•.

4

By Remark A.92, one has the following interpretation of the homologicaldimension.

Lemma A.94. If F• is an object of Db(X) and m ≥ 0 is a nonnegative integer,then hd(F•) ≤ m if and only if there is an integer j such that for any (closed)point x ∈ one has

HomiD(X)(F•,Ox) = 0 unless j ≤ i ≤ j +m.

The relationship between support and homological dimension is a conse-quence of some deep results in commutative algebra. One is the following acyclicitylemma of Peskine and Szpiro.

Lemma A.95. [246, Lemma 1.8], [107, Lemma 1.3] Let O be a local Noetherianring. Suppose that

0→M−s → · · · →M0 → 0

is a complex of O-modules whose cohomology modules H−i(M•) have finite length.If depthO(M−i) ≥ i for all 0 ≤ i ≤ s, then H−i(M•) = 0 for all i > 0. Moreover,depthO(M−i) > i for all 0 ≤ i ≤ s implies that H−i(M•) = 0 for all i.

Another important result is the so-called “new intersection theorem” [259](cf. also [107, Theorem 1.13] for a proof).

Theorem A.96. Let O be a local Noetherian ring of dimension d and m its maximalideal. Suppose that

0→M−s → · · · →M0 → 0

is a nonexact complex of free O-modules whose cohomology modules H−i(M•)have finite length. Then s ≥ d.

Corollary A.97. [71, Cor. 5.5] or [68, Cor. 5.2] Let F• be a nonzero object of Db(X).Then

codim(Supp(F•)) ≤ hd(F•) .


Proof. We can assume that F• is represented be a finite complex E• ≡ Em−s →· · · → Em of locally free sheaves, with s = hd(F•). If Y is an irreducible componentof Supp(F•) and y0 is the generic point of Y , by localization at y0 we obtain acomplex Em−sy0

→ · · · → Emy0of free modules over the local ring OX,y0 (the stalk

of OX at y0). The latter is a local Noetherian ring of dimension codim(Y ), andall the cohomology modules Hj(E•y0) ' (Hj(E•))y0 are supported on the uniqueclosed point y0 ∈ SpecOX,y0 . The result follows directly from Theorem A.96.

A refinement of the “new intersection theorem” has been proved by Bridge-land and Iyengar, who have fixed a gap in the proof of the similar statement [71,Thm. 4.3]. Let O be a local Noetherian ring of dimension d and m its maximalideal. Suppose that

0→M−s → · · · →M0 → 0

is a nonexact complex of free O-modules whose cohomology modules H−i(M•)have finite length. We now that s ≥ d by Theorem A.96.

Theorem A.98. [67, Thm. 1.1] Assume that O contains a field or dimO ≤ 3. Ifs = d and the residue field O/m is a direct summand of H0(M•), then O is regularand H−i(M•) = 0 for i > 0.

We can also characterize smooth schemes by means of the “new intersectiontheorem.”

Corollary A.99. [67, Cor. 1.2] Let Z be an irreducible algebraic variety and fix aclosed point x ∈ Z. Assume that there exists an object E•(x) in Db(Z) such thatfor any closed point z ∈ Z and any integer i, one has

HomiD(Z)(E•(x),Oz) = 0 unless z = x and 0 ≤ i ≤ dimZ.

Assume also that Ox is a direct summand of H0(E•(x)). Then Z is smoooth at xand E•(x) ' H0(E•(x)) in D(Z).

The proof of Corollary A.99 is analogous to that of the similar statement [71,Cor. 5.6] or [68, Cor. 5.3], by taking the care of replacing [71, Thm. 4.3] by [67,Thm. 1.1].

Appendix B

Lattices

In this appendix we gather together some results about integral lattices. In par-ticular, we state a counting formula, due to S. Hosono, B.H. Lian, K. Oguiso andS.-T. Yau [150], which is needed in Chapter 7 to compute the number of Fourier-Mukai partners of a K3 surface. Our basic references are classical monographssuch as [267] and [84], and Nikulin’s seminal paper [235].

B.1 Preliminaries

A lattice Λ is a free Z-module of finite rank equipped with an integral nonde-generate symmetric bilinear form 〈·, ·〉Λ : Λ × Λ → Z. Some comments about thisdefinition are perhaps not out of place. It is clear that, after having fixed a ba-sis e1, . . . , er, a lattice L can be regarded as a discrete subgroup of the realvector space Rr that generates all of it, i.e., as a “lattice,” according to Serre’sterminology. However, we would like to adopt here a more abstract point of view,since the main issue we are interested in is the classification of embeddings of onegiven lattice into another. For the same reason, the language of lattices we opt forseems to be more convenient than the (otherwise equivalent) language of integralquadratic forms.

Homomorphisms of lattices are homomorphisms of Z-modules preserving thebilinear form. An injective homomorphism is called an embedding and an isomor-phism an isometry. We denote by O(Λ) the group of isometries of Λ with itself.The direct sum of two lattices Λ1, Λ2 is the Z-module Λ1 ⊕ Λ2 endowed with thebilinear form 〈(x1, y1), (x2, y2)〉 = 〈x1, x2〉Λ1

+ 〈y1, y2〉Λ2.

The rank r(Λ) of the lattice Λ is its rank as a Z-module. If we fix a basise1, . . . , er, the determinant of the matrix 〈ei, ej〉 does not depend on the choice

340 Appendix B. Lattices

of this basis; it is called the discriminant of the lattice and denoted by d(Λ). Thesignature (τ+, τ−) of the lattice Λ is the signature of the R-extension of 〈·, ·〉Λ toΛ⊗ R. The integer τ(Λ) = τ+ − τ− is called the index of Λ.

We say that the lattice Λ is even (or of type II) if 〈λ, λ〉Λ ∈ 2Z for all λ ∈ Λ,and that is odd (or of type I) if it is not even. The lattice is called unimodular ifits discriminant is ±1.

Example B.1. For any integer n, we denote by I〈n〉 the rank 1 lattice generatedby a vector e such that 〈e, e〉 = n. The hyperbolic lattice U is the rank 2 (evenunimodular) lattice with a basis e1, e2 such that 〈e1, e1〉 = 〈e2, e2〉 = 0 and〈e1, e2〉 = 1. Another important example is provided by the rank 8 lattice whosebilinear form with respect to the canonical basis coincides with the Cartan matrixassociated to the exceptional Lie algebra e8, namely:

2 0 −1 0 0 0 0 00 2 0 −1 0 0 0 0−1 0 2 −1 0 0 0 0

0 −1 −1 2 −1 0 0 00 0 0 −1 2 −1 0 00 0 0 0 −1 2 −1 00 0 0 0 0 −1 2 −10 0 0 0 0 0 −1 2

. (B.1)

This lattice — which is denoted by E8 — is even, unimodular and positive definite.

4

Given a lattice Λ and an integer n, we denote by Λ〈n〉 the lattice obtainedby multiplying its bilinear form by n.

When its bilinear form is indefinite, a unimodular lattice is determined upto isomorphism by its rank, index and type. For a proof of the following classicalresult see, e.g., [267, Chap. V].

Theorem B.2. Let Λ be an indefinite unimodular lattice of rank r and index τ =τ+ − τ−. Then,

• if Λ is odd, it is isomorphic to the lattice (⊕τ+

1 I〈1〉)⊕ (⊕τ−1 I〈−1〉);

• if Λ is even and τ ≥ 0, it is isomorphic to the lattice (⊕p1U)⊕ (⊕q1E8), wherep = 1

2 (r − τ) and q = 18τ ;

• if Λ is even and τ < 0, it is isomorphic to the lattice (⊕p1U)⊕ (⊕q1E8〈−1〉),where p = 1

2 (r + τ) and q = −18τ .

Given two rational quadratic forms, the weak Hasse principle [267, Theorem9, Chap. IV] states that they are equivalent if and only if they are equivalent over

B.2. The discriminant group 341

Qp for all primes p and over R (here Qp is the field of p-adic numbers). Althoughthis is no longer true for integral quadratic forms (see [84, p. 129] for an example),the number of nonequivalent integral quadratic forms which are equivalent overZp for all primes p and over R is finite.

Let Λ be an even lattice. The genus g of Λ is defined as the set of isomorphismclasses of lattices Λ′ such that Λ⊗Zp ' Λ′⊗Zp for all primes p and Λ⊗R ' Λ′⊗R.

Theorem B.3. [84, Theorem 1.1, Chap. 9] The genus g is a finite set.

B.2 The discriminant group

Let Λ be a lattice. The dual Z-module Λ∗ = Hom(Λ,Z) is endowed with the dualQ-valued bilinear form 〈·, ·〉Λ∗ , which is naturally induced by 〈·, ·〉Λ. If we fix abasis for Λ and the dual basis for Λ∗, the matrix defining 〈·, ·〉Λ∗ is just the inversematrix of that defining 〈·, ·〉Λ. The bilinear form 〈·, ·〉Λ gives an immersion of Z-modules Λ → Λ∗; we shall identify Λ with its image in Λ∗. The restriction of〈·, ·〉Λ∗ to Λ coincides with 〈·, ·〉Λ.

Since Λ and Λ∗ are of the same dimension as Z-modules, their quotientAΛ = Λ∗/Λ is finite. Actually, the order of AΛ is equal to the absolute value ofthe discriminant of Λ. In particular, when Λ is unimodular, AΛ = 0, and indeedthe immersion Λ → Λ∗ is an isometry.

On AΛ is naturally defined a nondegenerate symmetric Q/Z-valued bilinearform bΛ : AΛ ×AΛ → Q/Z given by

bΛ([x], [y]) ≡ 〈x, y〉Λ∗ mod Z , (B.2)

for any [x], [y] in AΛ.

If Λ is even, the bilinear form bΛ : AΛ × AΛ → Q/Z induces a Q/2Z-valuedquadratic form qΛ on AΛ. One has

2bΛ([x], [y]) ≡ qΛ([x] + [y])− qΛ([x])− qΛ([y]) mod 2Z . (B.3)

We shall call the pair (AΛ, qΛ) the discriminant group associated with the latticeΛ. There is a canonical homomorphism O(Λ)→ O(AΛ, qΛ).

For example, it is an easy exercise to show that, for any integer n 6= 0, thediscriminant group of U〈n〉 is

AU〈n〉 = U〈n〉∗/U〈n〉 = Z/nZ⊕ Z/nZ

endowed with the quadratic form

qU〈n〉([x], [y]) ≡ 2nxy mod 2Z .


The following result shows that the genus of an even lattice is uniquely de-termined by its discriminant group and signature.

Theorem B.4. [235, Cor. 1.9.4] Two even lattices Λ, Λ′ of the same rank are inthe same genus if and only if τ(Λ) = τ(Λ′) and (AΛ, qΛ) ' (A′Λ, q

′Λ).

We shall make use of the following “stability criterion” in the sequel.

Theorem B.5. [235, Corollary 1.13.4] Let Λ be an even lattice having signature(τ+, τ−) and discriminant form q. The lattice Λ ⊕ U is the unique even latticehaving signature (τ+ + 1, τ− + 1) and discriminant form q.

Let Λ be an even lattice. A subgroup G ⊂ AΛ is called isotropic if qΛ|G = 0.Any isotropic group G determines an even lattice Σ together with an embeddingΛ → Σ such that the quotient Σ/Λ is the group G itself. Indeed, for any [x], [y]in G, the formula (B.3) shows that 2bΛ([x], [y]) ≡ 0 mod 2Z. Hence, the bilinearform bΛ induces a well-defined Z-valued even nondegenerate symmetric bilinearform on the Z-module Σ = x ∈ Λ∗|[x] ∈ G. It is clear that the image of Λ in Λ∗

is contained in Σ since G ⊂ AΛ. An even lattice Σ together with an embeddingΛ → Σ such that the quotient is a finite group is called an overlattice of Λ. Givensuch an overlattice, the natural immersion Λ → Λ∗ factors through the threeimmersions

Λ → Σ → Σ∗ → Λ∗ (B.4)

so that the group GΣ = Σ/Λ ⊂ Λ∗/Λ = AΛ turns out to be isotropic. So we haveproved the following useful result [235, Prop. 1.4.1].

Proposition B.6. Let Λ be an even lattice. There is a bijection between isotropicsubgroups GΣ of the discriminant group AΛ and overlattices Σ of Λ.

The quadratic form qΣ on the discriminant group AΣ can be obtained fromqΛ. By taking the quotient of (B.4) by Λ, we get the inclusions of Abelian groupsGΣ ⊂ Σ∗/Λ ⊂ AΛ. It is easy to check that Σ∗/Λ = G⊥Σ , where G⊥Σ is the orthogonalcomplement of GΣ in AΛ w.r.t. the bilinear form bΛ. Finally, G⊥Σ/GΣ = AΣ and

(qΛ|G⊥Σ)/GΣ = qΣ . (B.5)

B.3 Primitive embeddings

An embedding i : L → Σ of lattices is primitive if the quotient Σ/i(L) is a freeZ-module. Notice that this condition is equivalent to the requirement that theadjoint homomorphism i∗ : Σ∗ → L∗ is surjective. A vector x ∈ Σ is primitive ifthe sublattice generated by x is primitive.

B.3. Primitive embeddings 343

The orthogonal complement of a sublattice M → Σ in Σ is the sublattice

M⊥ = x ∈ Σ|〈x, y〉Σ = 0 for all y ∈M

endowed with the naturally induced bilinear form. It is straightforward to checkthat M⊥ ⊂ Σ is actually a primitive sublattice.

Let us consider a primitive embedding i : L → Σ, where both L and Σ areeven lattices, and a lattice K isomorphic to the orthogonal complement i(L)⊥ ofi(L) in Σ. Then L⊕K → Σ, and Σ is an overlattice of L⊕K. By Proposition B.6,this overlattice is uniquely determined by the isotropic subgroup GΣ = Σ/(L⊕K)of the discriminant group AL⊕K = AL ⊕ AK . The group GΣ is embedded inAL ⊕AK by the composition of homomorphisms

φΣ : GΣ = Σ/(L⊕K) → Σ∗/(L⊕K)→ (L⊕K)∗/(L⊕K) = AL ⊕AK . (B.6)

Using Equation (B.5), we can express the quadratic form qΣ on the discriminantgroup AΣ in the following way:

qΣ =((qL ⊕ qK)|G⊥Σ

)/GΣ . (B.7)

The composition of φΣ with the canonical projection p1 : AL ⊕ AK → AL yieldsa homomorphism φΣ,L : GΣ → AL, which is injective due to the primitivity of Lin Σ (actually, the injectivity of φΣ,L is equivalent to the primitivity condition).Analogously, we obtain an injective homomorphism φΣ,K = φΣ p2 : GΣ → AK .

When Σ is unimodular, by composing with the lattice isomorphism Σ ' Σ∗,we obtain a surjective homomorphism

Σ ' Σ∗ → L∗ (B.8)

and similarly for K∗. Hence, the maps φΣ,L, φΣ,K are isomorphisms. The compo-sition hΣ = φΣ,K φ−1

Σ,L : AL → AK satisfies the condition qK hΣ = −qL (theminus sign depends on the fact that GΣ is to be isotropic in AL ⊕ AK); in otherwords, hΣ is an isometry (AL, qL) ∼→ (AK ,−qK) [235, Prop. 1.6.1]. This isometryuniquely determines the primitive embedding i : L → Σ, where Σ is unimodularand the orthogonal complement of i(L) is isomorphic to K. We can rephrase thisresult in the following way.

Theorem B.7. Let L, K be even lattices. Any isometry h : (AL, qL)→ (AK ,−qK)uniquely determines an even, unimodular overlattice Σh of L ⊕K, together witha primitive embedding i : L → Σh whose orthogonal complement is isomorphic toK.

Let us now fix two even lattices L and Σ, assuming that Σ is unimodular.Fix a subgroup of isometries J ⊂ O(L). Two primitive embeddings i : L → Σ,


i′ : L → Σ are J-equivalent if there are isometries α ∈ O(Σ) and β ∈ J makingcommutative the following diagram:

Lβ //

_

i

L _

i′

Σ

α // Σ

When J reduces to the identity, we simply say that two embeddings are are equiv-alent.

We are interested in studying the set

EJ(L,Σ) = i : L → Σ |primitive embedding/J-equivalence

and in showing it is finite (we will follow the treatment given in [150]).

Given any two primitive embeddings i : L → Σ, i′ : L → Σ, let us considerK ' i(L)⊥ and K ′ ' i′(L)⊥. Both K and K ′ are even, and by Theorem B.7 onehas

(AK , qK) ' (AL,−qL) ' (AK′ , qK′) . (B.9)

Moreover, since Σ is an overlattice of both L⊕K and L⊕K ′, by tensoring by Rwe get isomorphisms of R-vector spaces Σ⊗R ' (L⊕K)⊗R ' (L⊕K ′)⊗R. So,τ(K) = τ(Σ)− τ(L) = τ(K ′). In view of Theorem B.4, we get that K and K ′ arein the same genus g. As we have noticed at the end of Section B.1, the genus is afinite set, so we let g = K1, . . . ,Ks, where the Ki’s are nonisometric lattices.

In some cases, the genus consists of just one element. The following result isa straightforward consequence of Theorem B.5.

Proposition B.8. Let i, i′ : L → Σ be primitive embeddings. Assume that both K

and K ′ contain the hyperbolic U lattice as a direct summand. Then K ' K ′, andthe two embeddings are equivalent.

On the other hand, if the primitive embeddings i : L → Σ, i′ : L → Σ are J-equivalent, we know that there are α ∈ O(Σ) and β ∈ J such that αi = i′β; thisimplies that the restriction α|i(L) : i(L)→ i′(L) is an isometry. As a consequence,α|i(L)⊥ : i(L)⊥ → i′(L)⊥ is an isometry as well. Hence, K ' K ′ ' Ki for someKi ∈ g.

Let E(L,Σ,Ki) be the subset of EJ(L,Σ) constituted by J-equivalence classesof primitive embeddings i : L → Σ such that i(L)⊥ ' Ki. By the previous discus-sion, it is clear that

EJ(L,Σ) =⋃Ki∈g

EJ(L,Σ,Ki) .

B.3. Primitive embeddings 345

Lemma B.9. The set EJ(L,Σ) is finite.

Proof. Let Oi the set of even unimodular overlattices of L ⊕Ki. The discussionbefore Theorem B.7 shows that there is a surjective map Oi → EJ(L,Σ,Ki). ByProposition B.6, Oi is finite.

More precisely, the cardinality of EJ(L,Σ) can be computed by a countingformula proved in [150].

Theorem B.10. In the previous notation, fix Ki ∈ g. One has

](EJ(L,Σ,Ki)) = ](O(Ki)\O(AKi)/J) .

Appendix C

Miscellaneous results

For the reader’s convenience, in this appendix we collect several standard resultsthat are used throughout this book. These concern relative duality, Simpson’snotion of stability for pure sheaves, and Fitting ideals.

C.1 Relative duality

We review here the foundations of relative duality. A more exhaustive treatmentmay be found in many standard references, see, e.g., [139, 291, 2, 234, 89].

Let f : X → Y be a proper morphism of algebraic varieties. The deriveddirect image functor Rf∗ is defined over the entire category D(X) and has a rightadjoint f ! : D(Y )→ D(X) (see Appendix A). There is a functorial isomorphism

HomD(Y )(RfX∗F•,G•) ' HomD(X)(F•, f !G•) . (C.1)

This also has a local form:

RHom•OY (Rf∗F•,G•) ' Rf∗RHom•OX (F•, f !G•) .

The complex f !OY is called the dualizing complex of f . The functor f ! is sometimesdetermined by the dualizing complex; indeed, whenever G• or f !OY is of finite Tor-dimension, there is an isomorphism

f !G• ' Lf∗G•L⊗ f !OY . (C.2)

Another important property of dualizing complexes is their compatibilitywith base change. If φ : T → Y is a morphism of finite Tor-dimension and fT : XT =

348 Appendix C. Miscellaneous results

X ×Y T → T , φX : XT → X are the induced morphisms, there is an isomorphism

Lφ∗Xf!G• ' f !

TLφ∗G• , (C.3)

provided that at least one of the morphisms fT and φX is flat (cf. Lipman’s articlein [2]).

A key feature is compatibility with the composition of morphisms. Assumethat in the next diagram all morphisms are proper:

X

g @@@@@@@@

f // Y

h

T

; (C.4)

since Rg∗ ' Rh∗ Rf∗ we have a functor isomorphism

g! ' f ! h! (C.5)

between the right adjoints. If one of the dualizing complexes h!OT or f !OY is offinite Tor-dimension, then (C.2) yields an isomorphism

g!OT ' Lf∗h!OTL⊗ f !OY . (C.6)

For particular morphisms we have more concrete expressions for the functorf ! and the dualizing complex. For example, whenever f is finite the isomorphism

f∗f!G• ' RHomOY (f∗OX ,G•) (C.7)

holds. This is the case, for instance, when f is a closed immersion. Assume thatin addition f is a local complete intersection of codimension d (in the sense of[119, 6.6]) defined by an ideal sheaf J ; that is, J is locally generated by a regularsequence of length d. Then all the cohomology sheaves of f !OY vanish but the d-thone, which turns out to be isomorphic to the normal bundle NY/X '

∧d(J /J 2)∗.So one has

f !OY ' NY/X [−d] 'd∧

(J /J 2)∗[−d] (C.8)

and

f !G• ' Lf∗G• ⊗d∧

(J /J 2)∗[−d] (C.9)

for every complex G• in D(X).

A Cohen-Macaulay morphism is a flat morphism whose fibers are Cohen-Macaulay varieties. When f is flat of relative dimension n, the condition thatf is Cohen-Macaulay is equivalent to the fact that all the cohomology sheaves

C.1. Relative duality 349

Hi(f !OY ) vanish for i 6= −n. In this case we call the sheaf ωX/Y = H−n(f !OY )the dualizing sheaf of f , and we have

f !OY ' ωX/Y [n] .

The relative dualizing complex can be also used to characterize Gorenstein mor-phisms, that is, flat morphisms whose fibers are Gorenstein varieties. A flat mor-phism of relative dimension n is Gorenstein if and only ifHi(f !OY ) = 0 for i 6= −n,(so that it is Cohen-Macualay) and the relative dualizing sheaf ωX/Y = Hi(f !OY )is a line bundle. The relative canonical divisors are the divisors KX/Y such thatωX/Y ' OX(KX/Y ). Smooth morphisms are Gorenstein and the relative dualizingsheaf in that case is the sheaf ωX/Y = ∧nΩX/Y of relative n-differentials.

When X is a proper Cohen-Macaulay variety X of dimension n and f is theprojection of X onto a point, Equation (C.1) yields

RΓ(X,F•)∗ ' RHomD(X)(F•, ωX [n]) or

Hi(X,F•)∗ ' Hom−iD(X)(F•, ωX [n]) ' Homn−i

D(X)(F•, ωX)

(C.10)

where Hi(X,F•) = Hi(RΓ(X,F•)) is the hypercohomology of the complex F• andωX is the dualizing sheaf of X. For a complex F concentrated in degree zero, oneobtains the usual Serre duality theorem for coherent sheaves on a Cohen-Macaulayproper variety [139, 141]

Hi(X,F) ' Extn−i(F , ωX)∗ .

If f : X → Y is a proper Gorenstein morphism, one has

f !G• ' f∗G• ⊗ ωX/Y [n] (C.11)

for G• in D(Y ). In this situation, the duality isomorphisms (C.1) and (C.10) takethe more familiar form

RHom•OY (Rf∗F•,G•) ' Rf∗RHom•OX (F•, f∗G• ⊗ ωX/Y [n])

HomD(Y )(RfX∗F•,G•) ' HomD(X)(F•, f∗G• ⊗ ωX/Y [n])(C.12)

for arbitrary complexes G• in D(Y ) and F• in D(X).

If X is a proper Gorenstein variety of dimension n, the isomorphism

HomD(X)(F•,G•) ' HomD(X)(G•F• ⊗ ωX [n])∗ (C.13)

holds when the complex F• has finite homological dimension; indeed in this casewe can use Proposition A.87, and since all Hom groups in (C.10) are finite-dimensional, by taking duals one has

HomD(X)(F•,G•) ' H0(RΓ(X,F•∨L⊗G•)) ' HomD(X)(F•∨

L⊗G•, ωX [n])∗

' HomD(X)(G•,F• ⊗ ωX [n])∗ .


In particular, given two coherent sheaves F and G, with F of finite Tor-dimension, there are duality isomorphisms

ExtiX(F ,G) ' Extn−iX (G,F ⊗ ωX)∗

where n = dimX and ωX is the dualizing line bundle.

We conclude this section with some results on adjunction. Let us considerproper morphisms f : X → Y , h : Y → T and g = h f : X → T .

Proposition C.1.

1. Assume that one of the two morphisms f , h is Gorenstein. If the other mor-phism is Cohen-Macaulay (resp. Gorenstein), then g is also Cohen-Macaulay(resp. Gorenstein) and

ωX/T ' Lf∗ωY/TL⊗ωX/Y .

2. If f and g are Gorenstein and h is flat and surjective, then h is Cohen-Macaulay and ωX/T ' f∗ωY/T ⊗ ωX/Y .

3. Assume that f is a closed immersion and a local complete intersection. If h isCohen-Macaulay (resp. Gorenstein), then g is Cohen-Macaulay (resp. Goren-stein) as well and

ωX/T ' f∗ωY/T ⊗NX/Y ,

where NX/Y is the normal bundle.

Proof. 1. If n, m are the relative dimensions of f and h, one has f !OY ' ωX/Y [n],h!OT ' ωY/T [m]. By Equation (C.6), one has g!OY ' f∗ωY/T ⊗ ωX/Y [m+ n].

2. Again by (C.6), ωX/T [m+n] ' f∗h!OT ⊗ωX/Y [n] and then f∗Hi(h!OT ) =0 for i 6= −m. Since f is flat and surjective it is faithfully flat, so thatHi(h!OT ) = 0for i 6= −m.

3. The proof is similar once we know that f !OY ' NY/X [−d], which followsfrom Equation (C.8).

In particular, if f is the immersion of a Cartier divisor Y in X, one has anisomorphism

ωY/T ' ωX/T |Y ⊗OY (Y )

and an induced linear equivalence of Cartier divisors

KY/T ≡ KX/T · Y + Y · Y (adjunction formula) .

C.2. Pure sheaves and Simpson stability 351

C.2 Pure sheaves and Simpson stability

A notion of stability for pure sheaves, generalizing the ordinary definition fortorsion-free sheaves on irreducible varieties, is due to Simpson [269].

Let X be a projective variety of dimension n, and fix a polarization H

in X. If E is a coherent sheaf on X, the Euler characteristic χ(X, E(sH)) =∑i≥0(−1)i dimHi(X, E(sH)) of the twisted sheaf E(sH) = E ⊗ OX(sH) can be

written as a polynomial in s with rational coefficients; it is called the Hilbertpolynomial of E , and has the form

P (E , s) = χ(X, E(sH)) =r(E)m!

sm +d(E)

(m− 1)!sm−1 + . . . (C.14)

where r(E) > 0 and d(E) are integer numbers and m ≤ n is the dimension of thesupport of E .

Definition C.2. [269] A coherent sheaf E is pure of dimension m if the supportof E has dimension m and the support of any nonzero subsheaf 0 → F → E hasdimension m as well. 4

When X is integral, the pure sheaves of dimension n = dimX are preciselythe torsion-free sheaves. We can then adopt the following definition.

Definition C.3. A coherent sheaf E is torsion-free if it is pure of dimension n =dimX. 4

Simpson also defined the reduced Hilbert polynomial and the (Simpson) slopeof E by the following formulas:

pS(E , s) =P (E , s)r(E)

, µS(E) =d(E)r(E)

.

This enables us to define (Simpson) Gieseker stability, µ-stability, Gieseker semista-bility and µ-semistability for pure sheaves as in the usual case.

Definition C.4. A pure sheaf E on X is µS-stable (resp. µS-semistable) with respectto H, if for every proper subsheaf F → E one has

µS(F) < µS(E) (resp. µS(F) ≤ µS(E)).

Analogously, a pure sheaf E on X is (Simpson) Gieseker-stable (resp. Gieseker-semistable) with respect to H, if for every proper subsheaf F → E there is aninteger s0 such that

pS(F , s) < pS(E , s) (resp. pS(F , s) ≤ pS(E , s))

for all s > s0. 4


If the sheaf E is of finite homological dimension, so that its Chern classesare defined, we can apply the Riemann-Roch theorem for singular varieties [119,Cor. 18.3.1] to prove that the Hilbert polynomial is given by

χ(X, E(s)) =∫

ch(E) · ch(OX(sH)) · Td(X) , (C.15)

where Td(X) is the Todd class of X; for a local complete intersection scheme, theclass Td(X) is the usual Todd class td(X) of the virtual tangent bundle. Since thedegree of the Hilbert polynomial of E is the dimension of the support of E , onehas

r(E) = chn−m(E) ·Hm ,

d(E) = (chn−m+1(E) + chn−m(E) · td1(X)) ·Hm−1 .(C.16)

If X is integral and E is torsion-free, the numbers r(E) and d(E) are closelyrelated to the usual rank rk(E) and degree deg(E) (with respect to H), because inthat case Equation (C.16) reads

r(E) = rk(E) · deg(X) , d(E) = deg(E) + rk(E)C (C.17)

where deg(X) is the degree of X defined in terms of H and C is a constant. Onethen sees that for a torsion-free sheaf E on an irreducible projective variety X,the Simpson notions of µS-stability and semistability (and Gieseker stability andsemistability) are equivalent to the usual ones. Equation (C.17) also suggests asensible definition of the rank of a sheaf on an arbitrary projective variety.

Definition C.5. Let X be a projective scheme and H polarization in X. The po-larized rank of a coherent sheaf E on X is the rational number

rk(X,H)(E) =r(E)

deg(X).

4

Note that if X is irreducible and Supp(E) is different from X the ordinaryrank of E is zero, but the polarized rank is not.

Simpson constructed moduli spaces for (Gieseker) stable and semistable sheaveson the fibers of a projective morphism by fixing the Hilbert polynomial (or reducedHilbert polynomial) of the sheaves. Let us state the relevant existence theorem[269, Theorem 1.21]. Let X → B be a projective morphism with a fixed rela-tive polarization H, P a rational polynomial and Mss

X/B,P the moduli functorof relatively pure semistable sheaves on the fibers (with respect to the inducedpolarization) which have Hilbert polynomial P . To be more precise, the functorMss

X/B,P associates to any variety f : T → B over B the set of equivalence classesof coherent sheaves E on T ×B X flat over T and whose restriction Et = j∗t E to

C.2. Pure sheaves and Simpson stability 353

the fiber Xf(t) is pure and semistable with respect to the induced polarization,and has Hilbert polynomial P . Here two such sheaves E and F are considered tobe equivalent if E ' F ⊗ π∗TL for a line bundle L on T .

We say that a morphism of schemes π : Mss(X/B,P )→ B is a coarse modulispace for the moduli functor Mss

X/B,P if there is a morphism of functors

φ : MssX/B,P → HomB( • ,Mss(X/B,P ))

which universally corepresents MssX/B,P . We recall that φ corepresents Mss

X/B,P ifthe following universal property holds: if Z → B is another B-scheme, for everymorphism of functors ψ : Mss

X/B,P → HomB( • , Z) there is a unique morphism ofB-schemes g : Mss(X/B,P )→ Z such that the diagram

MssX/B,P

φ //

ψ ))RRRRRRRRRRRRRRHomB( • ,Mss(X/B,P ))

g

HomB( • , Z)

commutes. The property that φ universally corepresents MssX/B,P means that for

every base change T → B the fiber product T ×B Mss(X/B,P ) corepresents thefiber product functor HomB( • , T ) ×HomB( • ,B) Mss

X/B,P . One easily sees that acoarse moduli space, if it exists, is unique up to isomorphisms in the category ofB-schemes.

Given a morphism T → B and a sheaf E on T ×B X flat over T , which isrelatively pure and semistable with Hilbert polynomial P , we denote by φE : T →Mss(X/B,P ) the induced morphism to the coarse moduli space.

We recall the definition of S-equivalence. As we shall see in Theorem C.6,this notion is needed to ensure the existence a coarse moduli space of semistablesheaves with prescribed topological invariants (actually, points in the moduli spaceparameterize S-equivalence classes of semistable sheaves rather than semistablesheaves themselves). This definition requires the notion of Jordan-Holder filtration:every semistable sheaf F has a filtration F = Fm ⊃ Fm−1 ⊃ · · · ⊃ F0 = 0whose quotients Fi/Fi−1 are stable with the same slope as F . The Jordan-Holderfiltration is not unique but the associated graded sheaf G(F) = ⊕iFi/Fi−1 isuniquely determined. Two semistable sheaves F , G are then called S-equivalent ifG(F) ' G(G). Thus two stable sheaves are S-equivalent if and only if they areisomorphic.

Theorem C.6.

1. There exists a coarse moduli scheme π : Mss(X/B,P ) → B for the mod-uli functor Mss

X/B,P of relatively semistable pure sheaves with fixed Hilbertpolynomial P .


2. The morphism π : Mss(X/B,P )→ B is projective.

3. The closed points of the fiber Mss(Xt, p) = π−1(t) of π : Mss(X/B,P ) →B over a closed point t ∈ B represent S-equivalence classes of semistablesheaves on the fiber Xt with Hilbert polynomial P .

4. There exists an open subscheme Ms(X/B,P ) ⊆Mss(X/B,P ) whose closedpoints represent the isomorphism classes of stable sheaves on the fibers ofX → B.

5. Locally for the etale topology on Ms(X/B,P ), there exists a universal sheafEuniv on Ms(X/B,P )×B X →Ms(X/B,P ).

If a universal sheaf exists globally (and not only locally) on Ms(X/B,P )×BX →Ms(X/B,P ), we say that Ms(X/B,P ) is a fine moduli space, a terminologythat we have already used in some parts of this book. In this case, Ms(X/B,P )→B represents the moduli functor Ms

X/B,P of relatively stable sheaves, namely,the morphism of functors φ : Mss

X/B,P → HomB( • ,Mss(X/B,P )) induces anisomorphism

φ : MsX/B,P

∼→ HomB( • ,Ms(X/B,P )) .

This implies that, given a morphism T → B and sheaf E on T ×B X flat overT , which is relatively pure and stable with Hilbert polynomial P , then there is aunique morphism φE : T →Ms(X/B,P ) such that

E ' (φE × IdX)∗Euniv ⊗ π∗TL

for a line bundle L on T .

In the general situation, a universal sheaf Euniv only exists locally in the etaletopology (also in the analytic topology, if the base field is the field of the complexnumbers). This means that there exist an etale covering γ : M →Ms(X/B,P ) anda sheaf Euniv on M ×B X → M , flat over M and relatively pure and stable withHilbert polynomial P , fulfilling the following universal property: take a morphismT → X and a sheaf E on T×BX flat over T , which is relatively pure and stable withHilbert polynomial P . We then have an induced morphism φE : T →Ms(X/B,P ),which enables us to consider the fiber product T = T ×Ms(X/B,P ) M and theinduced etale covering γ × IdX : T ×B X → X. One has an isomorphism

(γ × IdX)∗E ' (φE × IdX)∗Euniv ⊗ π∗TL

of sheaves on T ×T X for a line bundle L on T . In words, though E is not thepullback of the universal sheaf, it is so “locally in the etale topology,” that is,it becomes the pullback of the universal sheaf after a base change by an etalecovering. Analogous properties hold true for Gieseker semistability.

C.3. Fitting ideals 355

The following result is very useful. Let P (s) = rs + d a rational polynomialof degree 1, where r and d are integer numbers, r > 0. We consider relatively puresemistable sheaves with Hilbert polynomial P on the fibers of a projective mor-phism X → B. This means that the relevant sheaves on the fibers are supportedon curves.

Proposition C.7. If r and d are coprime, every pure semistable sheaf on a fiber Xt

is stable and Ms(X/B,P )→ B is a projective morphism and a fine moduli space,so that there exists a universal family Euniv on Ms(X/B,P )×BX →Ms(X/B,P ).

Proof. Assume E is a semistable sheaf on a fiber Xt with Hilbert polynomialχ(Xt, E(sHt)) = rs+ d (here Ht is the polarization induced by H on the fiber). IfE is not stable, there is an exact sequence

0→ F → E → G → 0

of coherent sheaves on Xt such that µS(F) = µS(E) = d/r. The sheaf E is pure ofdimension one, so that F is pure of dimension one too and its Hilbert polynomialis of the form r(F)s + d(F). Since Hilbert polynomials are additive and r(F) ispositive, we have r(F) ≤ r. From r(F)d = rd(F) and the coprimality of r and d

we obtain r(F) = r and d(F) = d. Thus the Hilbert polynomial of G is zero, sothat G = 0, proving that E is stable. The existence of a universal family can beseen using the arguments of [227, Theorem A.6].

C.3 Fitting ideals

We review here some elementary properties of the Fitting ideals. Further informa-tion can be found, for instance, in [258]. Let A be a ring that we assume Noetherianfor simplicity. If M is a finitely generated A-module, we can think of M as thecokernel of a morphism between free modules of finite rank,

L1φ−→ L0 →M → 0 .

This is called a finite presentation of M . If we choose bases of L0 and L1, themorphism φ is determined by a n×m matrix (aij) where n and m are the ranksof L0 and L1.

Definition C.8. The i-th Fitting ideal of M is the ideal Fi(M) of A generated bythe minors of order n− i of the matrix (aij). 4

Fitting ideals are well defined, in the sense that they depend only on themodule M and not on the matrix expression of a finite presentation φ, nor on thechoice of the finite presentation itself. We now list a few more properties of the


Fitting ideals. The first is that the ring A/Fi(M) is compatible with base change, inthe following sense: if A→ B is a ring morphism, one has Fi(M⊗AB) = Fi(M)·B,so that

B/Fi(M ⊗A B) ' A/Fi(M)⊗A B . (C.18)

Since A/Fi(M) is the ring corresponding to the closed subscheme of SpecA de-fined by the Fitting ideal Fi(M), Equation (C.18) means that this subscheme iscompatible with base change.

The second fact is that Fitting ideals are multiplicative over direct sums ofmodules [258, 5.1], that is,

Fi(M ⊕N) =∑h+j=i

Fh(M) · Fj(N) ,

and in particularF0(M ⊕N) = F0(M) · F0(N) . (C.19)

Though in general the Fitting ideals are not multiplicative over exact se-quences, the following property is true: if

0→M → P → N → 0

is an exact sequence of modules, then

F0(P ) ⊆ F0(M) · F0(N) . (C.20)

Fitting ideals can be defined for coherent sheaves F on algebraic varieties X, sincethe local Fitting ideals Fi(F(U)) constructed for the OX(U)-modules F(U) forevery affine open subset U ⊆ X coincide on the intersections, thus gluing togetherto give an ideal sheaf Fi(F).

Let us denote by Zi(F) the closed subscheme defined by the Fitting idealFi(F). Since the 0-th Fitting ideal F0(F) is contained in the annihilator of F , onehas Z0(F) ⊇ Supp(F). These two closed subschemes are very similar: they havethe same isolated components, though these can be counted more times in Z0(F)than in the support, while the embedded components may be different. We thendefine:

Definition C.9. The modified support of F is the closed subscheme

Supp0(F) = Z0(F) ⊇ Supp(F)

defined by 0-th Fitting ideal F0(F). 4

The base change property (C.18) now gives

C.3. Fitting ideals 357

Lemma C.10. Let p : X → B a morphism of algebraic varieties and F a coherentsheaf on X. For every point s ∈ B, the restriction Zi(F)s = Zi(F) ∩ Xs to thefiber Xs of the closed subscheme Zi(F) defined by the Fitting ideal Fi(F) is theclosed subscheme Zi(Fs) defined by the Fitting ideal Fi(Fs) of the restriction Fsof F to the fiber,

Zi(F)s ' Zi(Fs) .

In particular, the modified support Supp0(F) is compatible with base changes, i.e.,

Supp0(Fs) ' Supp0(F)s = Supp0(F) ∩Xs .

The latter property justifies the introduction of the modified support, sincethe ordinary support Supp(F) does not enjoy this property.

Our next aim is to compute the cohomology class of the modified supportSupp0(F) = Z0(F) of a coherent sheaf. We start with a description of the 0-thFitting ideal of a coherent sheaf F in terms of a presentation of F as the cokernelof a morphism between locally free sheaves of finite rank

E1φ−→ E0 → F → 0 .

One has an exact sequence

s∧E1 ⊗ det E−1

0

∧sφ⊗1−−−−→ OX → OZ0(F) → 0 (C.21)

where s = rk E0.

Proposition C.11. Let X be a smooth projective variety X of dimension n > 0 andF a torsion coherent sheaf on it, that is, rk(F) = 0. Then one has

detOZ0(F) ' detF ,

so that c1(OZ0(F)) = c1(F).

Proof. Let us write F as a cokernel E1φ−→ E0 → F → 0 as above and let N be the

image of φ, so that there is an exact sequence

0→ N τ−→ E0 → F → 0 . (C.22)

Now (C.21) implies that the sequence

n∧N ⊗ (det E0)−1 ∧nτ⊗1−−−−→ OX → OZ0(F) → 0 (C.23)

is exact. The sheaf N has rank s = rk(E0) as rk(F) = 0 and is locally free in thecomplementary U of the n−1-singularity set Sn−1(N ). The exact sequence (C.21)


implies that Sn−1(N ) ⊂ Sn−2(F), so that Sn−1(N ) has codimension at least 2(cf. for instance [144, Prop. 1.13], or [184, Thm. 5.8] in the complex case). Moreover∧sN ⊗ (det E0)−1

|U is a line bundle and∧sN ⊗ (det E0)−1

|U∧sτ⊗1−−−−→ F0(F)|U is

an isomorphism by [140, Theorem 3.8]). Thus detF0(F) ' det(∧sN⊗(det E0)−1).

Moreover,∧sN coincides with

∧s(N )∗∗ ' detN on U , and then det(F0(F)) 'detN ⊗ (det E0)−1. By the multiplicativity of the determinant bundle

detF ' det E0 ⊗ (detN )−1 ' (det(F0(F)))−1 ' det(OZ0(F)) .

Proposition C.12. Let E be a coherent sheaf on a smooth projective variety X withSupp(E) of codimension 1. The polarized rank of E as a sheaf on the modifiedsupport Supp0(E) of E (Definitions C.5 and C.9) is one,

rk(Supp0(E),H)(E) = 1 .

Moreover, if the polarized rank of E as a sheaf on its ordinary support is also one,then the modified support coincides with the ordinary one, Supp0(E) = Supp(E).

Proof. If m = dimX, then the numerator of the leading coefficient of the Hilbertpolynomial (Eq. (C.14)) is

r(E) = c1(E) ·Hm−1 = c1(OZ0(E)) ·Hm−1

= [OZ0(E)] ·Hm−1 = [Supp0(E)] ·Hm−1

by Equation (C.16), where the second equality is due to Proposition C.11. Thisproves the first part by Definition C.5. For the second, if rk(Supp(E),H)(E) = 1, thendeg(Supp0(E)) = deg(Supp(E)) and the closed immersion Supp(E) → Supp0(E) isan isomorphism.

This result deserves a comment, since the polarized rank of a sheaf E on itsordinary support Y = Supp(E) equals the rank (understood as the 0-th Cherncharacter) of the restriction E|Y because of Equation (C.16). Take for instance anintegral Cartier divisor j : Y → X, a locally free sheaf F of rank r on X andlet E = F ⊗ j∗OY . Then Y = Supp(E) and the rank of E on Y is r. However ifY0 = Supp0(E) is the modified support, the polarized rank rkY0,H(E) is one, thatis, E has rank one on Y0. This is not contradictory because the modified supportof E is Y0 = rY ; to see this, notice that

0→ F(−Y )φ−→ F → E → 0

is a finite presentation of E as the quotient of a morphism of locally free OX -modules. If f is a local equation of Y , we see that locally the matrix of φ is ftimes the identity r× r matrix. Then the 0-th Fitting ideal is locally (fr) so thatY0 = rY as claimed.

Appendix D

Stability conditions for derived

categories

by Emanuele Macrı

D.1 Introduction

The notion of stability condition on a triangulated category has been introducedby Bridgeland in [65], following ideas from physics by Douglas [104] on π-stabilityfor D-branes. A stability condition on a triangulated category T is given by ab-stracting the usual properties of µ-stability for sheaves on complex projectivevarieties; one introduces the notion of slope, using a group homomorphism fromthe Grothendieck group K(T) of T to C, and requires that a stability conditionhas generalized Harder-Narasimhan filtrations and is compatible with the shiftfunctor. The main property is that there exists a parameter space Stab(T) for sta-bility conditions, endowed with a natural topology, which is a (possibly infinite-dimensional) complex manifold. The space of stability conditions Stab(T) thusyields a geometric invariant naturally attached to a triangulated category T.

For motivations and interpretations of stability conditions from the physicsviewpoint we refer the reader to the original papers by Douglas, Aspinwall, andothers (see, e.g., [104, 103, 9, 58] and references therein). Here we shall concentrateon the mathematical aspects of the definition. From the mathematical viewpoint,one of the main motivations for introducing stability conditions on a triangulatedcategory T is to single out subsets of objects of T which can be classified viasome sort of “well-behaved” moduli space. The basic example to keep in mind isBridgeland’s construction [62] of the three-dimensional flop as a moduli space of

360 Appendix D. Stability conditions for derived categories

“stable” objects in the derived category; we already met this in Section 7.5. Moregenerally, the idea is that fixing a stability condition provides the data necessaryto reconstruct a variety from its bounded derived category.

Another motivation comes from the fact that the space of stability conditionsgives a geometric object which is useful in studying algebraic structures, e.g.,t-structures and groups of autoequivalences. More precisely, hard combinatorialquestions (like the structure of spherical objects on the derived category of a K3surface) can be reduced to more manageable geometric questions. In this direction,the conjecture stated in [66], and explained in Section D.3.1 of this appendix, isthe main example.

Unfortunately it is not easy to construct examples of stability conditions.In particular, up to now, there is no example of stability conditions on derivedcategories of smooth and projective Calabi-Yau threefolds: hence Bridgeland’sconstruction of the three-dimensional flop cannot be interpreted as a moduli spaceof stable objects. At the same time, stability conditions on the derived categoryof the local model of the three-dimensional flop can be described and the flopinterpreted as a moduli space of stable point-like objects (see Section 7.5 and[282]).

The following cases have been so far studied, at different levels of detail:

• smooth projective curves [65, 205, 241];

• singular elliptic curves [79];

• local resolutions of surface singularities of ADE type [278, 59, 161, 240]. Inparticular in [161] the case of singularities of type A has been completelydescribed (for the A1 case see also [240, 206]);

• smooth and projective K3 and Abelian surfaces [66] (for Abelian surfacesand generic twisted K3 surfaces see also [157]);

• Enriques surfaces (or more generally equivariant derived categories) [253,206];

• projective spaces and Del Pezzo surfaces [12, 204];

• the total space of the canonical bundle over the projective plane [64] (forcanonical bundles over Del Pezzo surfaces see also [38]);

• local three-dimensional crepant resolutions [282];

• local fibrations in elliptic curves or in K3 and Abelian surfaces [283];

• some graded matrix factorizations arising from regular weight systems [275,170];

D.1. Introduction 361

• generic analytic tori is in [216].

A subject which is receiving a growing interest is the definition of invariantsof a (say smooth and projective) variety X from moduli spaces of “stable” ob-jects in the derived category of X (see, for example, [169, 167, 72, 244, 245, 280,31, 279, 172]). In a series of papers (see in particular [169]), Joyce has starteda program for studying invariants constructed in this way and, in particular, forunderstanding how they vary under a change of the stability condition. The ideais that these invariants should be encoded in holomorphic generating functionsthat are globally well defined on the space of stability conditions (for an interpre-tation in terms of families of isomonodromic irregular connections on P1 see [72]).This part of the theory is still largely conjectural, starting from the choice of thedefinition of stability condition. Some clarifications (at least for the Calabi-Yauthree-dimensional case) are provided in [172].

For the case of K3 surfaces the situation is somehow more clear, especiallyafter Toda’s paper [281] (see Section D.4 of this appendix), where moduli space ofsemistable objects with respect to Bridgeland’s stability conditions are constructed(at least as Artin stacks of finite type), and a conjecture by Joyce on how invariantsvary is completely solved.

These various conjectural aspects show how the theory of stability conditionsis still at its very beginning, and quite likely, the most interesting developments arestill to be discovered. The definition itself might need to be modified to some ex-tent, at least in the non-Calabi-Yau case. Some other definitions have been indeedproposed after Bridgeland. In [128] a definition is given by modeling Rudakov’sstability for Abelian categories (see [261]), but in that generality the constructionof the space of stability conditions is missing. In [31, 279] a sort of “limit” ofBridgeland stability conditions is defined, which is quite useful for constructingexamples in any dimension. A more refined definition of Bridgeland stability con-dition is in [172]. Finally, in [159] a definition of stability is given having in mindmostly the construction of “good” moduli space of stable objects.

In this appendix we shall explain in some detail the basic properties of stabil-ity conditions. In Section D.2 we shall essentially follow Bridgeland’s foundationalpaper [65]. In the first part, we shall give the definition of stability condition andstate Bridgeland’s main theorem (Theorem D.8) on the existence of a naturaltopology on the space of stability condition giving it the structure of a complexmanifold. Then we will examine the easiest examples in which the space of sta-bility condition can be explicitly described: bounded derived categories of smoothprojective curves. Finally, in the last part, we shall sketch the proof of TheoremD.8.

In Section D.3 we shall describe — following [66] — an important exam-ple, in which the space of stability conditions can be (at least partly) described:


the bounded derived category of a smooth projective K3 surface. After statingBridgeland’s result (Theorem D.19), we shall outline its proof, first by construct-ing stability conditions on K3 surfaces, proving a covering property for certainconnected components of the space of stability conditions and then, after recallingthe important wall and chamber structure, singling out a particular connectedcomponent. All the results of this section hold also for Abelian surfaces. Actuallyin this case Conjecture D.20 can be proved to be true (see [243, 66, 157]).

In Section D.4 we shall consider the question of constructing moduli spacesof “semistable” objects and shall have a first glimpse at counting invariants forderived categories of K3 surfaces, following [281]. The main results of this sectionare Theorem D.35, which says that, after fixing a stability condition and numericalinvariants, the set of semistable objects of a given phase is an Artin stack of finitetype over C, and Theorem D.45, which is the solution of a conjecture by Joyceon how certain invariants of semistable objects vary on the space of stabilityconditions.

A word on notation. All our schemes and varieties will be always over thecomplex numbers, and most categories will be linear over C. Derived functors willoften be denoted as their underived counterparts (i.e., we shall often omit R andL in front of them).

This appendix is a survey of the main ideas that have been developed aboutstability conditions by a number of authors. Among other survey papers existingin the literature we shall cite [12] for an introduction from the physical viewpointand [63, 58] for an account of Bridgeland’s work, more examples, explanations,and further directions of study.

Acknowledgments

The structure and presentation of this appendix has been inspired by a series oflectures given by Sukhendu Mehrotra, Paolo Stellari, and Yukinobu Toda at the“First CTS Conference on Vector Bundles” at the Tata Institute for FundamentalResearch in Mumbai. I would like to thank them for their beautiful lectures andthe organizers of the conference for making this possible. Many thanks are dueto Stefano Guerra and Paolo Stellari for comments, suggestions, and for goingthrough a preliminary version of the manuscript.

D.2 Bridgeland’s stability conditions

In this section we outline Bridgeland’s paper [65]. In the first part we give thedefinition of stability condition on a triangulated category and state Bridgeland’smain theorem (Theorem D.8) showing that stability conditions are parameterized

D.2. Bridgeland’s stability conditions 363

by a (possibly infinite-dimensional) manifold. Then we show the first examples:stability conditions on the derived categories of curves, following [65, 205, 241].Finally we present the proof of Theorem D.8.

D.2.1 Definition and Bridgeland’s theorem

Let T be an essentially small triangulated category (i.e., T is equivalent to a smallcategory, i.e., a category whose class of objects is a set). Denote by K(T) theGrothendieck group of T.

Definition D.1. A stability condition on T is a pair σ = (Z,P) where Z : K(T)→ Cis a group homomorphism (the central charge) and P(φ) ⊂ T are full additivesubcategories, φ ∈ R, satisfying the following conditions:

1. If 0 6= E ∈ P(φ), then Z(E) = m(E) exp(iπφ) for some m(E) ∈ R>0.

2. P(φ+ 1) = P(φ)[1] for all φ ∈ R.

3. If φ1 > φ2 and Ei ∈ P(φi) (i = 1, 2), then HomT(E1, E2) = 0.

4. Any 0 6= E ∈ T admits a Harder-Narasimhan filtration (HN-filtration forshort) given by a collection of distinguished triangles Ei−1 → Ei → Ai →Ei−1[1] with E0 = 0 and En = E such that Ai ∈ P(φi) with φ1 > . . . > φn.

4

The nonzero objects in the category P(φ) are said to be (σ-)semistable ofphase φ, while the objects Ai in (4) are the semistable factors of E. Note thatan HN-filtration of a nonzero object E is unique up to a unique isomorphism. Wewrite φ+

σ (E) = φ1, φ−σ (E) = φn, and mσ(E) =∑j |Z(Aj)|.

For any interval I ⊂ R, one defines P(I) as the extension-closed subcategoryof T generated by the subcategories P(φ), for φ ∈ I. Note that the definitionis consistent when I is the interval consisting of only one point, i.e., P(φ) isextension-closed.

Remark D.2. (i) For all φ ∈ R, P((φ, φ+ 1]) is the heart of a bounded t-structureon T (we gave the notion of heart of a t-structure in Definition 7.54). This isshown quite easily by considering the subcategory P(> φ) ⊂ T and using theHN-filtrations to find its right orthogonal in T. The category P((0, 1]) is called theheart of σ.

(ii) As a corollary of (i), each subcategory P(φ) is Abelian. The simple objectsof P(φ) are called (σ-)stable of phase φ. 4

Let A be a small Abelian category. A stability function on A is a grouphomomorphism Z : K(A)→ C such that Z(E) ∈ H for all 0 6= E ∈ A, where H =


0 6= z ∈ C : z/|z| = exp(iπφ), with 0 < φ ≤ 1. Given a stability function Z, thephase of a nonzero object E ∈ A is defined to be φ(E) = (1/π) argZ(E) ∈ (0, 1].An object 0 6= E ∈ A is called semistable (with respect to Z) if φ(A) ≤ φ(E) for allnonzero subobjects A → E. A stability function is said to have the HN-propertyif every nonzero object of A admits a finite (Harder-Narasimhan) filtration withsemistable quotients of decreasing phases.

Using Remark D.2, it is easy to see that a stability condition on a triangulatedcategory T induces a stability function on the Abelian category P((0, 1]) for allφ ∈ R. The existence of HN-filtrations implies that the induced stability functionhas the HN-property, the semistable objects of phase φ being precisely the nonzeroobjects of P(φ), for φ ∈ (0, 1]. An important feature of Bridgeland’s stabilityconditions is that the converse of this is also true.

Proposition D.3. Giving a stability condition on T is equivalent to giving a boundedt-structure on T and a stability function on its heart with the HN-property.

Proof. Let A be the heart of a bounded t-structure on T and let Z : K(A) → Cbe a stability function with the HN-property. By using the fact that K(A) canbe identified with K(T), we can define a central charge Z : K(T) → C. Definenow P(φ), for each φ ∈ (0, 1], as the subcategory of T consisting of the semistableobjects of A with phase φ, together with the zero object. For φ ∈ R, set P(φ) =P(ψ)[k], where ψ ∈ (0, 1], k ∈ Z, and φ = ψ + k. It is not difficult to check thatthe pair (Z,P) thus defined yields a stability condition on T.

In view of this result, sometimes we shall denote a stability condition as apair (Z,A), where A is the heart of a bounded t-structure and Z is a stabilityfunction on it having the HN-property.

Example D.4. Let C be a smooth projective curve over C, let Coh(C) be thecategory of coherent sheaves on it, and Db(C) = Db(Coh(C)) its bounded derivedcategory. Define Z : K(C)→ C by Z(E•) = −deg(E•)+i rk(E•) for all E• ∈ K(C).It is easy to see that this defines a stability function on Coh(C), whose semistableobjects are either µ-semistable vector bundles or torsion sheaves on C. The HN-property follows from the existence of Harder-Narasimhan filtrations for sheaves.By Proposition D.3 this induces a stability condition on Db(C). 4Remark D.5. (i) If A ⊂ T is the heart of a bounded t-structure and moreoverit is an Abelian category of finite length (i.e., Artinian and Noetherian), then agroup homomorphism Z : K(A)→ C with Z(S) ∈ H, for all simple objects S ∈ A,extends to a unique stability condition on T. This follows from the fact that A isgenerated by taking extensions of its simple objects.

(ii) Let A be a finite-dimensional associative algebra over C and let Db(A) =Db(Mod-A) the bounded derived category of (right) finitely generated A-modules.


As Mod-A is of finite length, by using the previous remark one can constructexamples of stability conditions on Db(A). 4

By using Remark D.2 one can prove (see [65, Lemma 4.3]) that, given astability condition (Z,P) on a triangulated category T, for all intervals I of lengthless than 1, the subcategory P(I) ⊂ T is quasi-Abelian, the strict exact sequencesbeing triangles of T all of whose vertices are in P(I) (see Definition A.10).

A stability condition is called locally finite if there exists some ε > 0 suchthat for all φ ∈ R each quasi-Abelian subcategory P((φ − ε, φ + ε)) is of finitelength. In this case P(φ) has finite length as well and so every object in P(φ) hasa finite Jordan-Holder filtration into stable factors of the same phase. The set oflocally finite stability conditions will be denoted by Stab(T).

Example D.6. (i) If C a smooth projective curve over C, the stability conditionconstructed in Example D.4 is locally finite. To show this one uses the fact thatthe image of the central charge is a discrete subgroup of C.

(ii) Let A be a finite-dimensional associative algebra over C. Any of thestability conditions constructed in Remark D.5 is locally finite. This follows fromthe fact that P((0, 1]) = Mod-A is of finite length. 4

We want to define a topology on Stab(T). Let σ = (Z,P) ∈ Stab(T). Definea map ‖ − ‖σ : HomZ(K(T),C)→ [0,+∞] by letting

‖U‖σ = sup|U(E)||Z(E)|

: E is σ-semistable.

Moreover, for τ = (W,Q) ∈ Stab(T) we define

f(σ, τ) = sup06=E∈T

|φ+σ (E)− φ+

τ (E)|, |φ−σ (E)− φ−τ (E)|∈ [0,+∞].

We may note that f depends only on the slicings P and Q; indeed f defines ageneralized metric on the set of slicings, i.e., it has all the properties of a metricbut it may be infinite (see [65, Sect. 6] for definitions and details). Finally, forε ∈ (0, 1/4), define

Bε(σ) = τ = (W,Q) : ‖W − Z‖σ < sin(πε) and f(σ, τ) < ε ⊂ Stab(T).

Lemma D.7. The set Bε(σ) : σ ∈ Stab(T), ε ∈ (0, 1/4) yields a basis for atopology on Stab(T).

Proof. We have to show that given ε1, ε2 ∈ (0, 1/4) and σ1, σ2 ∈ Stab(T), for allτ ∈ Bε1(σ1) ∩Bε2(σ2) there exists an η > 0 such that Bη(τ) ⊂ Bε1(σ1) ∩Bε2(σ2).

From the definition, it will be sufficient to prove that, given ε ∈ (0, 1/4) andσ ∈ Stab(T) such that τ ∈ Bε(σ), there exists an η > 0 for which Bη(τ) ⊂ Bε(σ).


This follows easily from the following inequality

k1‖U‖σ < ‖U‖τ < k2‖U‖σ (D.1)

for some constants k1, k2 > 0 and for all U ∈ HomZ(K(T,C). We leave the proofof (D.1) to the reader (for details see [65, Lemma 6.2]).

We endow Stab(T) with the topology generated by the basis of open sub-sets Bε(σ). By [65, Prop. 8.1], this topology can be equivalently described as thetopology induced by the generalized metric

d(σ1, σ2) =

sup06=E∈T

|φ+σ2

(E)− φ+σ1

(E)|, |φ−σ2(E)− φ−σ1

(E)|,∣∣∣∣log

mσ2(E)mσ1(E)

∣∣∣∣ ∈ [0,∞], (D.2)

for σ1, σ2 ∈ Stab(T).

Let now Σ ⊂ Stab(T) be a connected component. By (D.1) the subspace

U ∈ HomZ(K(T),C) : ‖U‖σ < +∞ ⊂ HomZ(K(T),C)

is locally constant on Stab(T) and hence constant on Σ. Denote it by V (Σ). Notethat ‖−‖σ, for σ ∈ Σ, defines a norm on V Σ. Moreover, by (D.1), all norms ‖−‖σon V (Σ) are equivalent, and therefore they define the same topology on V (Σ). Byits own definition, the map Z : Σ→ V (Σ) which associates to a stability conditionits central charge is continuous. Bridgeland’s main theorem asserts that this mapis actually a local homeomorphism.

Theorem D.8. (Bridgeland) For all connected components Σ ⊂ Stab(T), the mapZ : Σ→ V (Σ) which associates to a stability condition its central charge is a localhomeomorphism. In particular, Σ has a manifold structure, locally modeled on thetopological vector space V (Σ).

We shall sketch a proof of this theorem in Section D.2.3.

If K(T)⊗C is finite-dimensional over C, then Stab(T) is a finite-dimensionalcomplex manifold. This is the case, for example, when T = Db(A) for a finite-dimensional associative algebra A over C. On the contrary, a smooth projectivecurve of positive genus has infinite-dimensional Grothendieck group. In general,to get spaces parameterizing stability conditions that are finite dimensional com-plex manifolds, it is sufficient to take finite-dimensional slices of HomZ(K(T),C).An example, which is well suited for derived categories of smooth projective vari-eties, is obtained by asking that the central charge factorizes through the singularcohomology of X. This can be formalized as follows.


Suppose that the category T is C-linear and of finite type, that is, for any pairof objects E and F the space ⊕iHomT(E,F [i]) is a finite-dimensional C-vectorspace. The Euler-Poincare form on K(T)

χ(E,F ) =∑i∈Z

(−1)i dimC HomT(E,F [i])

allows us to define the numerical Grothendieck group N (T) = K(T)/K(T)⊥

(where orthogonality is with respect to χ). If N (T) has finite rank, then T issaid to be numerically finite. For example, T = Db(X) = Db(Coh(X)), for Xa smooth projective variety over C, is numerically finite by the Riemann-Rochtheorem.

A stability condition σ = (Z,P) such that Z factors through the epimorphismK(T)→ N (T) is called numerical. We denote by StabN (T) the set of locally finitenumerical stability conditions. An immediate consequence of Theorem D.8 is thatif T is numerically finite, then StabN (T) becomes a finite-dimensional complexmanifold, locally modeled over V (Σ) ∩HomZ(N (T),C).

The next result provides an important property of the space of stabilityconditions.

Proposition D.9. The space of stability conditions Stab(T) carries a right action

of the group Gl+

2 (R), the universal cover of Gl+2 (R), and a left action of the groupAut(T) of exact autoequivalences of T. These two actions commute.

Proof. The group Gl+

2 (R) acts in the following way. Consider a pair (G, f), withG ∈ Gl+2 (R) while f : R→ R is an increasing map, such that f(φ+ 1) = f(φ) + 1and G exp(iπφ)/|G exp(iπφ)| = exp(2iπf(φ)), for all φ ∈ R. It is easy to see that

Gl+

2 (R) can be thought as the set of such a pairs. Then (G, f) maps (Z,P) ∈Stab(T) to (G−1 Z,P f), where P f(φ) = P(f(φ)).

For the second action, Φ ∈ Aut(T) maps (Z,P) to (Z φ−1,Φ(P)), where φis the automorphism of K(T) induced by Φ.

Note that the action of Aut(T) preserves the generalized metric d of (D.2),i.e., Aut(T) acts on Stab(T) by isometries. Clearly, a result analogous to Proposi-tion D.9 holds for StabN (T) when T is numerically finite. We shall see in SectionD.3 that this can be applied to obtain information on the group of autoequiv-alences of derived categories of smooth projective varieties from the topology ofStabN (Db(X)).

We conclude this section by spending a few words on the moduli problem,which could be roughly summarized in the following two questions:


(1) given a (numerical) stability condition σ on T, does there exist a good notionof “moduli space” of σ-semistable objects?

(2) If (1) is true, then how do moduli spaces vary under changes of the stabilitycondition?

In Section D.4 we shall examine these two questions when T = Db(X), forX a smooth and projective K3 surface over C. Here we consider a simple examplewhere the answer to these questions is quite straightforward.

Example D.10. Let A be a finite-dimensional associative algebra over C and letDb(A) be as in Remark D.5(ii). Then A = Mod-A is an Abelian category of finitelength with a finite number of simple objects S1, . . . , Sn. As we saw in RemarkD.5, given z1, . . . , zn ∈ H, we can define a stability condition σ with heart A andstability function Z(Si) = zi, for all i, where K(A) = K(A) ' ⊕iZ[Si]. In thiscase, an object V of Db(A) is (semi)stable with respect to σ if and only if it is ashift of a θ-(semi)stable A-module in the sense of King [182], where θ is definedby

θ(U) = −=(Z(U)Z(V )

)∈ R,

for all U ∈ K(A). (We shall denote by < and = the real and imaginary partof a complex number.) In particular, by the results in [182], one can constructmoduli spaces Mv(σ) of (S-equivalence classes of) semistable objects in A havingfixed class v ∈ K(A) using geometric invariant theory techniques. Mv(σ) is thena projective variety over C. This gives a complete answer to question (1) for thisexample. 4Example D.11. In the situation of Example D.10, take A = C[Q] as the path-algebra associated to the quiver Q : • → • with two vertices and one arrow fromthe first to the second vertex. Then

A = Mod-A ' Rep(Q) =(V1, V2, φ) :

Vi finite-dimensional C-vector spacesφ : V1 → V2 a linear map

.

There are only three indecomposable objects in A: the two simple objects S1 =(C, 0, 0) and S2 = (0,C, 0) and their unique nontrivial extension E = (C,C, Id),0→ S2 → E → S1 → 0. One may show that, fixing v = [S1] + [S2], one has

Mv(σ) '

Spec(C) if arg(Z(S2)) ≤ arg(Z(S1)),0 if arg(Z(S2)) > arg(Z(S1)).

(This follows from the fact that S2 is the only proper subobject of E.) This givesan example related to question (2) above. 4


D.2.2 An example: stability conditions on curves

We give here some examples of spaces of stability conditions. We consider the caseof smooth projective curves over C of positive genus, where stability conditionsare substantially equivalent to µ-stability for sheaves. Then, for completeness, wesketch the case of P1. The results of this section, motivated by open problems in anearly version of [65], are contained in [65, 205, 241]. A study of stability conditionson singular elliptic curves, not included here, is in [79].

Let C be a smooth projective curve over C of positive genus. We will denoteby Stab(C) the space of locally finite numerical stability conditions on Db(C).The Riemann-Roch theorem shows that N (C) = N (Db(C)) can be identifiedwith Z⊕ Z, with the quotient map K(C)→ N (C) sending a class E• ∈ K(C) tothe pair consisting of its rank and degree.

In Example D.4 we constructed a stability condition on Db(C) which is nu-merical and locally finite (see Example D.6(i)). We want to prove that, up to the

action of Gl+

2 (R), this is the only one ([205, Thm. 2.7], however the case of ellipticcurves had been previously treated by Bridgeland).

Proposition D.12. Let C be a smooth projective curve over C of genus g(C) ≥ 1.

The action of Gl+

2 (R) on Stab(C) is free and transitive, so that

Stab(C) ' Gl+

2 (R) ' H× C,

where H ⊂ C is the complex upper half plane.

Proof. First of all recall the following technical fact (see [128, Lemma 7.2]): (*)If E ∈ Coh(C) is included in a triangle F• → E → G•, with F•,G• ∈ Db(C) andHom≤0

Db(C)(F•,G•) = 0, then F•,G• ∈ Coh(C).

Now it is not difficult to prove that the skyscraper sheaf Ox, x ∈ C, is stablein all stability conditions in Stab(C). Indeed, an easy consequence of (*) is that Oxis semistable and moreover all its stable factors are isomorphic to a single objectK• ∈ Db(C). But this implies that K• ∈ Coh(C) and so that K• ' Ox. In the sameway it can be shown that all line bundles are stable in all stability conditions.

Take σ = (Z,P) ∈ Stab(C) and a line bundle L on C. By what we haveseen above, L and Ox are stable in σ with phases φL and ψx respectively. Theexistence of maps L → Ox and Ox → L[1] gives inequalities ψx − 1 ≤ φL ≤ ψx,which implies that if Z is an isomorphism (seen as a map from H∗(C,R) ' R2 toC ' R2) then it must be orientation preserving. But Z is an isomorphism: indeedif not, there exist stable objects with the same phase having nontrivial morphisms,which is impossible. Hence, acting by an element of Gl

+

2 (R), one can assume thatZ(E•) = −deg(E•) + i rk(E•) and that for some x ∈ C, the skyscraper sheaf Ox


has phase 1. Then all line bundles on C are stable in σ with phases in the interval(0, 1) and, as a consequence, all skyscraper sheaves are stable of phase 1. But thisimplies that P((0, 1]) = Coh(C) so that the stability condition σ is precisely theone induced by µ-stability on C.

Remark D.13. Two remarks on the previous proposition are in order. Let us notethat although the stability conditions on a curve of positive genus are all in thesame Gl

+

2 (R) orbit, their hearts are not at all trivial. For example, in the caseof elliptic curves, it is possible to prove that the choice of a stability condition isequivalent to the choice of a noncommutative structure on C in the sense of Pol-ishchuk and Schwarz [254, 252]: the heart of a stability condition corresponds to theAbelian category of vector bundles with respect a noncommutative structure on C.Secondly, already in the case of elliptic curves, the quotient Stab(C)/Aut(Db(C))is of some interest. Indeed it can be proved (e.g., from the study of their actionon Stab(C)) that the autoequivalences of Db(C) are generated by shifts, automor-phisms of C and twists by line bundles together with the Fourier-Mukai transformassociated to the Poincare sheaf (see [228]). Automorphisms of C and twists byline bundles of degree zero act trivially on Stab(C) and one obtains

Stab(C)/Aut(Db(C)) ' Gl+2 (R)/Sl2(Z),

which is a C∗-bundle over the moduli space of elliptic curves. 4

The case of the projective line over C is slightly more involved, due to thepresence of “degenerate” stability conditions, i.e., stability conditions with veryfew stable objects. This purports some evidence that the definition of stabilitycondition for categories with nontrivial Serre functor may need some modification.

The basic idea for studying Stab(P1) = Stab(Db(P1)) ' StabN (Db(P1)) is touse the well-known Beılinson equivalence Db(P1) ' Db(A), where A is the pathalgebra associated to the Kronecker quiver • 2−→ • consisting of two vertices andtwo arrows from the first to the second vertex. In this way, examples of stabilityconditions can be constructed using both Example D.4 and Remark D.5(ii). In

this case the action of Gl+

2 (R) is neither free nor transitive. Nevertheless, we can

look at the subgroup C → Gl+

2 (R) given by z = x + iy 7→ (exp(x)Ry, fy), whereRy ∈ Gl+2 (R) is the rotation by the angle −πy and fy(φ) = φ+ y, for φ ∈ R. Thisaction of C on Stab(P1) is free and the quotient Stab(P1)/C is isomorphic to C(see [241, Sect. 4]). Hence we deduce the following result [241, Sec. 4].

Proposition D.14. Stab(P1) ' C2.

The fact that Stab(P1) is connected and simply connected was proved inde-pendently in [205].


D.2.3 Bridgeland’s deformation lemma

In this section we sketch a proof of Theorem D.8. We need to show that, for allconnected components Σ ⊂ Stab(T), the continuous map Z : Σ → V (Σ) whichassociates to a stability condition its central charge is a local homeomorphism.For more details see [65, Sect. 7].

We start with a useful lemma which shows that Z is locally injective.

Lemma D.15. Let σ = (Z,P), τ = (Z,Q) ∈ Stab(T) be stability conditions withthe same central charge Z such that f(σ, τ) < 1. Then σ = τ .

Proof. Suppose that σ 6= τ . There exists E ∈ P(φ) such that E /∈ Q(φ). Sincef(σ, τ) < 1, there is a triangle A → E → B such that A ∈ Q((φ, φ + 1)) andB ∈ Q((φ − 1, φ]). We claim that both A and B are nonzero. Indeed, assumeA = 0. Then E ∈ Q((φ − 1, φ]), which is a contradiction since σ and τ have thesame central charge. In the same way, B 6= 0.

Now, by the same argument, A /∈ P(≤ φ). Hence there exists an objectC ∈ P(ψ), with ψ > φ and a nonzero morphism C

h−→ A. Since E ∈ P(φ),the composite map C

h−→ A → E must be zero, and so h factorizes throughC

g−→ B[−1] → A. But, since f(σ, τ) < 1, C ∈ Q((ψ − 1, ψ + 1)) ⊂ Q(> (φ − 1))and B[−1] ∈ Q((φ− 2, φ− 1]), and so g = 0, a contradiction.

To conclude the proof of the theorem we need to show the following defor-mation lemma, due to Bridgeland.

Lemma D.16. Let σ ∈ Stab(T). Fix ε0 > 0 be such that ε0 < 1/10 and, for allφ ∈ R, each of the quasi-Abelian categories P((φ − 4ε0, φ + 4ε0)) ⊂ T is of finitelength. If 0 < ε < ε0 and W : K(T)→ C is a group homomorphism satisfying

|W (E)− Z(E)| < sin(πε)|Z(E)| (D.3)

for all σ-stable E ∈ T, there is a locally finite stability condition τ = (W,Q) on T

with f(σ, τ) < ε.

Notice that, by Lemma D.15, the stability condition τ is unique. Moreover,asking that Equation (D.3) is satisfied for all σ-stable objects in T clearly implies,by local finiteness, that it is satisfied by all σ-semistable objects. Hence TheoremD.8 follows.

To prove Bridgeland’s deformation lemma D.16 we need some preparatoryresults. Let W : K(T)→ C and ε be as in the statement. Call a subcategory of T

thin if it is of the form P((a, b)), for a and b real numbers such that 0 < b− a <1− 2ε. A thin subcategory is quasi-Abelian.


By Equation (D.16), if E ∈ T is σ-semistable, the phases of W (E) andZ(E) differ at most by ε. Hence, if A = P((a, b)) is thin, W defines a skewedstability function on A, that is a group homomorphism W : K(A) → K(T) → C(the Grothendieck group of A being the quotient of the free group generated bythe isomorphism classes of the objects of A with respect of strict short exactsequences) such that W (E) ∈ Ha,b for all objects 0 6= E ∈ A, where Ha,b = 0 6=z ∈ C : z/|z| = exp(iπψ), with a − ε < ψ < b + ε. Exactly in the same way asfor Abelian categories, we can define a notion of semistability, by looking at thephase ψ(−) = (1/π) argW (−) ∈ (a − ε, b + ε) and at the strict subobjects. Anobject of A which is semistable with respect to this skewed stability function willbe called W -semistable.

Example D.17. Suppose E is W -semistable in some thin subcategory A ⊂ T andset ψ = ψ(E). Then E ∈ P((ψ − ε, ψ + ε)). To show this, one uses the fact thatthe phases of the points W (A) and Z(A) differ by at most ε, for A ∈ T semistablein σ. 4

Let A be a thin subcategory of T. A nonzero object E ∈ A is said to beenveloped by A if a + ε ≤ ψ(E) ≤ b − ε. It can be checked (see [65, Lemma 7.5])that, if E ∈ T is enveloped by two thin subcategories B and C, then E is W -semistable in B if and only if it is W -semistable in C. More precisely, the notionof W -semistability for enveloped object is independent of the choice of the thinsubcategory containing it.

For all ψ ∈ R, defineQ(ψ) ⊂ T as the full additive subcategory of T consistingof the zero object together with those E ∈ T that are W -semistable of phaseψ in some thin enveloping subcategory P((a, b)). The pair τ = (W,Q) definesa stability condition on T. Indeed, the conditions (1) and (2) of Definition D.1are automatically satisfied by definition. Condition (3) can be proved by usingExample D.17 (see [65, Lemma 7.6]).

To prove the existence of HN-filtrations is more delicate. First of all it canbe shown (see [65, Lemma 7.7]) that if A = P((a, b)) ⊂ T is a thin subcategory offinite length, every nonzero object of P((a+ 2ε, b− 4ε)) has a finite HN-filtration,whose factors are W -semistable objects of A that are enveloped by A. Using thisand Example D.17 it is not difficult to show that, if for any t ∈ R we define asubcategory Q(> t) as the full extension-closed subcategory of T generated by thesubcategories Q(ψ), for ψ > t, then Q(> t) is a bounded t-structure on T. Definesimilarly Q(I) for any interval I ⊂ R.

Let 0 6= E ∈ T. We need to construct a finite filtration by objects of thecategories Q(ψ). Since Q(> t) is a bounded t-structure, we can assume E ∈Q((t, t+ 1]), for some t ∈ R. But using the t-structure Q(> (t+ δ)) we can further

D.3. Stability conditions on K3 surfaces 373

assume E ∈ Q((t, t+ δ]), for δ = 5ε0 − 5ε fixed. But now

Q((t, t+ δ]) ⊂ P((t− ε, t+ ε+ δ)) ⊂P((t− 3ε, t+ 5ε+ δ)) ⊂ P((t− 3ε0, t+ 5ε0)).

Since P((t− 3ε0, t+ 5ε0)) is thin and of finite length, by [65, Lemma 7.7] we haveHN-filtrations.

Finally, Example D.17 shows that f(σ, τ) < ε. Hence τ is locally finite since,for all t ∈ R, one has Q((t − ε, t + ε)) ⊂ P((t − 2ε, t + 2ε)). This concludes theproof of Lemma D.16.

Remark D.18. Let X be a smooth projective variety over C. Set Stab(X) =StabN (X) as the set of locally finite numerical stability conditions on Db(X).

(i) Let σ = (Z,P) be a numerical stability condition such that the image ofZ is a discrete subgroup of C (i.e., Z is discrete). Fix some 0 < ε < 1/2. Then,for all φ ∈ R, the quasi-Abelian category P((φ − ε, φ + ε)) is of finite length. Inparticular σ is locally finite. This follows from the fact that for a given objectE ∈ P((φ−ε, φ+ε)) the central charges of all sub- and quotient objects of E lie ina certain bounded region, and from the discreteness assumption. See also RemarkD.5.

(ii) A connected component Σ ⊂ Stab(X) is called full if the subspace V (Σ)∩HomZ(N (T),C) is equal to HomZ(N (T),C). A stability condition σ ∈ Stab(X) iscalled full if it belongs to a full connected component. Take σ = (Z,P) ∈ Stab(X)a full stability condition and fix 0 < ε < 1/2. Then, for all φ ∈ R, the quasi-Abeliancategory P((φ − ε, φ + ε)) is of finite length. Indeed, Theorem D.8 and the factthat σ is full allow one to find a stability condition τ = (W,Q) ∈ Stab(X) withdiscrete central charge such that f(σ, τ) < η, for η > 0 sufficiently small. Thenone uses (i) and the fact that P((φ− ε, φ+ ε)) ⊂ Q((φ− ε− η, φ+ ε+ η)). 4

D.3 Stability conditions on K3 surfaces

The aim of this section is to describe in detail a connected component of thespace of (numerical, locally finite) stability conditions on smooth and projectiveK3 surfaces, following [66].

After stating Theorem D.19, we sketch a proof and construct examples ofstability conditions on K3 surfaces. As an interlude, we introduce the wall andchamber structure, which will be very important in Section D.4 to study howmoduli spaces of semistable objects behave under a change of the stability condi-tion.


D.3.1 Bridgeland’s theorem

Let X be a projective K3 surface over the complex numbers and denote byStab(X) = StabN (Db(X)) the space of numerical locally finite stability condi-tions on Db(X). Recall that the numerical Grothendieck group in this case isisomorphic to the lattice

H1,1(X,Z) = H0(X,Z)⊕NS(X)⊕H4(X,Z) ⊂ H∗(X,Z),

where NS(X) is the Neron-Severi group of X (see Section 4.1). For a Z-module R,we set H1,1(X,Z)R = H1,1(X,Z)⊗Z R and NS(X)R = NS(X)⊗Z R.

By Orlov’s representability theorem 2.15, every autoequivalence Φ: Db(X) ∼→Db(X) induces a Hodge isometry fΦ : H(X,Z) ∼→ H(X,Z) (see Section 4.2). Letσ = (Z,P) ∈ Stab(X). Since π(σ) is in H1,1(X,Z)C, we can write Z(E•) =〈π(σ), v(E•)〉, for all E• ∈ Db(X), where v(E•) = ch(E•) ·

√td(X) is the Mukai

vector of E•, and 〈·, ·〉 is the Mukai pairing (see Section 1.1). By Theorem D.8, weget a continuous map π : Stab(X)→ H1,1(X,Z)C. Denote by P(X) the subset ofH1,1(X,Z)C defined asw1 + iw2 ∈ H1,1(X,Z)C :

w1, w2 ∈ H1,1(X,Z)R are linearly independent(aw1 + bw2)2 > 0 for all a, b ∈ R, (a, b) 6= (0, 0)

It is easy to see that P(X) has two connected components, P+(X) andP−(X), which are exchanged by conjugation. P+(X) is defined as the connectedcomponent containing the vector (1, iω,−ω2/2), for ω ∈ H1,1(X,Z) the class ofan ample divisor. Furthermore, set ∆(X) = δ ∈ H1,1(X,Z) : δ2 = −2 and, forδ ∈ ∆,

δ⊥ = Ω ∈ H1,1(X,Z)C : 〈Ω, δ〉 = 0.

Theorem D.19. There is a connected component Stab†(X) ⊂ Stab(X) which ismapped by π onto the open subset

P+0 (X) = P+(X) \

⋃δ∈∆(X)

δ⊥ ⊂ NS(X)C.

Moreover, the induced map π : Stab†(X) → P+0 (X) is a covering map and the

group

Aut†0(Db(X)) =

Φ ∈ Aut(Db(X)) :

Φ(Stab†(X)) = Stab†(X)

fΦ = Id : H(X,Z)→ H(X,Z)

acts freely on Stab†(X) and is the group of deck transformation of π.


Before starting the proof of Theorem D.19, we observe that a more detailedtopological study of the connected component Stab†(X) would yield a descriptionof the group of autoequivalences of the derived category of a K3 surface. Indeed,as remarked in [66], Theorem D.19 is not enough to determine the structure ofAut(Db(X)). Bridgeland conjectured the following.

Conjecture D.20. The action of Aut(Db(X)) on Stab(X) preserves the connectedcomponent Stab†(X). Moreover Stab†(X) is simply connected. 4

Denote by O(H(X,Z)) the group of Hodge isometries of H(X,Z) and byO+(H(X,Z)) the subgroup consisting of the isometries which preserve the orien-tation of the positive four-space in H(X,R). The following conjecture is due toBridgeland and Szendroi [66, 273].

Conjecture D.21. There is a short exact sequence of groups

1→ π1(P+0 (X))→ Aut(Db(X))

f(−)

−−−→ O+(H(X,Z))→ 0 . (D.4)

4

By Theorem D.19, a proof of Conjecture D.20 would imply Conjecture D.21.The fact that every autoequivalence of Db(X) induces an orientation-preservingHodge isometry in cohomology was conjectured by Szendroi in [273], by seeing it asa “mirror-symmetric” version of a result of Donaldson [101, 55] about the orienta-tion preserving property for diffeomorphisms of K3 surfaces. Szendroi’s conjecturehas been proved in [157, 156], by demonstrating Conjectures D.20 and D.21 inthe easier case of analytic K3 surfaces with trivial Picard group and then usingthe deformation theory of Fourier-Mukai kernels. At this point, the surjectivity inEquation (D.4) follows easily from the theory of moduli spaces on K3 surfaces.

Theorem D.22. The morphism f (−) : Aut(Db(X)) → O(H(X,Z)) induces a sur-jective morphism f (−) : Aut(Db(X))→ O+(H(X,Z)).

D.3.2 Construction of stability conditions

In this section we show that Stab(X) is nonempty by exhibiting a collection ofstability conditions σβ,ω on X. We also describe some special semistable objectsin σβ,ω. We begin by observing that the category of coherent sheaves cannot bethe heart of a stability condition on a variety Y if the dimension of Y is greaterthan 1 (the proof below is taken from [279, Lemma 2.7]).

Proposition D.23. Let Y be a smooth projective variety over C of dimension d ≥ 2.There is no numerical stability condition σ ∈ Stab(Y ) with heart Coh(Y ).


Proof. Assume that there exists σ = (Z,P) ∈ Stab(Y ) with heart Coh(Y ). WriteZ(E) =

∑dj=0(uj + ivj) · chj(E), for all E ∈ Coh(Y ), where uj , vj ∈ H2d−2j(Y,R)

and chj(E) ∈ H2j(Y,Q) is the j-th component of the Chern character of E . Sinced ≥ 2, there exists a smooth subvariety S ι−→ Y of dimension 2. The compositionK(S) ι∗−→ K(Y ) Z−→ C induces a numerical stability function on Coh(S). Hence, wecan assume d = 2.

Let C ⊂ Y be a smooth curve and take a divisor D on C. Then, by assump-tion, we have

=(Z(OC(D))) = v2(deg(D) + ch2(OC)) + v1 · [C] ≥ 0.

Since D can be of arbitrary degree, we must have v2 = 0. Similarly, using OY (mC)for m sufficiently small, we have v1 · [C] = 0. Therefore =(Z(OC(D))) = 0, and so

<(Z(OC(D))) = u2(deg(D) + ch2(OC)) + u1 · [C] ≤ 0.

By repeating the same argument, we have u2 = 0. But then Z(Ox) = u2 + iv2 = 0,for all skyscraper sheaves Oy, y ∈ Y , a contradiction.

So it is clear that to find an example of a stability condition on a K3 surface Xwe need first to indentify a good heart of a t-structure. Let β, ω ∈ NS(X)R with ωan ample R-divisor (we write ω ∈ Amp(X)). Moreover, let Zβ,ω : H1,1(X,Z)→ Cbe the morphism induced by

bZβ,ω(E•) = (exp(β + iω), v(E•))M , for all E• ∈ Db(X),

where exp(β + iω) = 1 + β + iω + 12 (β2 − ω2 + 2iβ · ω), and furthermore define

Tβ,ω =

E ∈ Coh(X) :

either E = Etor orµ−ω (E/Etor) > β · ω

,

Fβ,ω =E ∈ Coh(X) : E torsion-free and µ+

ω (E) ≤ β · ω,

Aβ,ω =

E• ∈ Db(X) :Hi(E•) = 0 if i /∈ −1, 0H−1(E•) ∈ Fβ,ωH0(E•) ∈ Tβ,ω

,

where µ−ω (resp. µ+ω ) denotes the smallest (resp. largest) slope of the Harder-

Narasimhan filtration of a torsion-free sheaf on X with respect to µω-stability andEtor denotes the torsion part of a sheaf E ∈ Coh(X).

By [137, Prop. 2.1] and the properties of µ-stability, Aβ,ω is the heart of abounded t-structure on Db(X). Let us note that since the canonical bundle of aK3 surface is trivial, an object E• ∈ Db(X) is spherical if HomDb(X)(E•, E•[i]) ' Cwhen i = 0, 2 and HomDb(X)(E•, E•[i]) = 0 for all other indices (cf. Section 2.4).


Lemma D.24. The morphism Z = Zβ,ω defines a stability function on Aβ,ω ifand only if Z(E) /∈ R≤0, for all E ∈ Coh(X) spherical. In particular, this holds ifω2 > 2.

Proof. The only case we need to check is when E ∈ Fβ,ω and =(Z(E)) = (c− rβ) ·ω = 0, where we have set v(E) = (r, c, s). We have to prove that <(Z(E)) > 0.

By taking a filtration with respect to µω-stability, we can assume that E isµω-stable. Then, by the Riemann-Roch theorem, we have c2 − 2rs = v(E)2 ≥ −2,with equality if and only if E is spherical. Moreover, by the Hodge index theorem,we have (c− rβ)2 ≤ 0. Hence, writing <(Z(E)) explicitly, we have

<(Z(E)) =12r((c2 − 2rs) + rω2 − (c− rβ)2

).

The claim now is clear.

Lemma D.25. Assume that β, ω ∈ NS(X)Q and ω ∈ Amp(X) are such thatZβ,ω(E) /∈ R≤0 for all spherical E ∈ Coh(X). The stability function Zβ,ω onAβ,ω has the HN property, hence it defines a numerical stability condition σβ,ωon Db(X). Moreover, this stability condition is locally finite.

Proof. We only delineate the main argument. Set φ(−) = (1/π) arg(Zβ,ω(−)) ∈[0, 1). As in the classical existence results for Harder-Narasimhan filtrations (see[261, Thm. 2] for a general approach, and [65, Prop. 2.4] for this case), for Zβ,ωto have the HN property the following two conditions are to be satisfied:

(a) There are no infinite chains of subobjects in Aβ,ω

. . . ⊂ E•j+1 ⊂ E•j ⊂ . . . ⊂ E•2 ⊂ E•1

with φ(E•j+1) > φ(E•j), for all j.

(b) There are no infinite chains of quotients in Aβ,ω

E•1 E•2 . . . E•j E•j+1 . . .

with φ(E•j) > φ(E•j+1), for all j.

Assume for a contradiction that an infinite chain as in (a) does exist. Since=(Zβ,ω) is discrete, then there exists a positive integer N ∈ N such that 0 ≥=(Zβ,ω(E•n)) = =(Zβ,ω(E•n+1)), for all n ≥ N . Let F•n = E•n/E•n+1 ∈ Aβ,ω.Then, by additivity, =(Zβ,ω(Fn)) = 0, for all n ≥ N . Hence φ(F•n) = 1, for alln ≥ N and φ(E•n) ≥ φ(E•n+1, a contradiction.

Assume now that an infinite chain as in (b) exists. As before, 0 ≥ =(Zβ,ω(E•n))= =(Zβ,ω(E•n+1)), for all n ≥ N . Let G•n = ker(E•N E•n) ∈ Aβ,ω. Then=(G•n) = 0 and φ(G•n) = 1 for all n ≥ N .


We need the following facts.

(i) Let P(1) be the full Abelian subcategory of Aβ,ω whose objects are thoseP• ∈ Aβ,ω having =(Zβ,ω(P•)) = 0. Then P(1) is of finite length. To show this,one uses the fact that the real part of Zβ,ω is discrete.

(ii) By using this, and the explicit description of objects in Aβ,ω, one mayshow that for all such objects E• there exists an exact sequence 0→ A• → E• →B• → 0 in Aβ,ω such that A• ∈ P(1) and HomAβ,ω (P•,B•) = 0 for all P ∈ P(1).

Then there exists an exact sequence 0→ A• → E•N → B• → 0 in Aβ,ω suchthat A• ∈ P(1) and HomAβ,ω (P•,B•) = 0, for all P• ∈ P(1). Since G•n ∈ P(1)then the inclusion G•n ⊂ E•N factorizes through G•n ⊂ A• ⊂ E•N . Hence we getan infinite chain of subobjects of A•

0 = G•N ⊂ G•N+1 ⊂ . . . ⊂ A•,

which contradicts fact (i) above.

The fact that σβ,ω is locally finite follows from Remark D.18(i) since Zβ,ω isdiscrete.

By Lemma D.25, σβ,ω is in Stab(X) if β and ω are rational. We shall see thatHN filtrations and the local finiteness hold also for general real β, ω, provided thecondition of Lemma D.24 is satisfied.

Now we examine some particular semistable objects in σβ,ω.

Example D.26. All skyscraper sheaves Ox, for x ∈ X are stable in σβ,ω of phase1, for β and ω as in Lemma D.25. This follows from the fact that they are simplein Aβ,ω. 4

Proposition D.27. (“large volume limit”) Assume β = 0 and ω ∈ NS(X) ample.Let E• ∈ Db(X) be such that r > 0 and c · ω > 0, where v(E•) = (r, c, s). Then E•is semistable in σn = (Z0,nω,A0,nω) for a sufficiently large n ∈ N if and only if E•is a shift of a ω-Gieseker semistable (torsion-free) sheaf on X.

Proof. Observe that A0,nω = A0,ω = A and that

Zn(r, c, s) = Z0,nω(r, c, s) = (rn2ω2

2− s) + inc · ω.

Assume E• σn-semistable for all n 0. Up to shift, we can assume E• ∈ A.The first step is to show that E• = E ∈ T0,ω and that it is torsion-free (henceµ−ω (E) > 0). For this, consider the exact sequence in A

0→ H−1(E•)[1]→ E• → H0(E•)→ 0,


with H−1(E•) ∈ F0,ω and H0(E•) ∈ T0,ω. Now H−1(E•) being torsion-free impliesthat

limn→∞

(1/π) arg(Zn(H−1(E•)[1])) = 1 > 0 = limn→∞

φn(E•).

Hence E• is not σn-semistable for n 0 unless H−1(E•) = 0. So, E• = E ∈ T0,ω.The proof that E is torsion-free is analogous.

For the second step, assume that E is not ω-Gieseker semistable. Then thereexists an exact sequence in Coh(X)

0→ G → E → B → 0,

with G Gieseker semistable destabilizing E . Note that, since µ−ω (E) > 0, G,B ∈A. Writing explicitly the condition that E is σn-semistable for n 0 gives, forv(G) = (r(G), c(G), s(G)) and v(E) = (r(E), c(E), s(E)),

n2ω2

2r(G)r(E)

(c(E) · ωr(E)

− c(G) · ωr(G)

)+ (s(E)(c(G) · ω)− s(G)(c(E) · ω)) > 0

for n 0. As a consequence, either µω(G) < µω(E) or µω(G) = µω(E) and

s(E)r(E)

≥ s(G)r(G)

.

Both cases lead to a contradiction.

For the converse implication, assume E ∈ Coh(X) torsion-free and ω-Giesekersemistable. Let 0→ A• → E → B• → 0 be an exact sequence in A. Then we havean exact sequence in Coh(X)

0→ H−1(B•)→ A• ' H0(A•)→ E → H0(B•)→ 0.

Hence µω(A•) ≤ µ+ω (A•) ≤ µω(E). We use now the following boundedness result

(see [66, Lemma 14.3]): if F ∈ Coh(X) is a torsion-free ω-Gieseker semistable sheafwith µω(F) > 0, there exists an integer P ∈ N such that, for all proper subobjects0 6= G• → F in A, we have s(G•)/r(G•) ≤ P .

After setting φn(−) = (1/π) arg(Zn(−)) ∈ [0, 1) from this we have, for all n,

φn(E)− φn(A•) = n

[n2ω

2

2(r(A•)(c(E) · ω)− r(E)(c(A•) · ω))

+r(A•)r(E)(c(A•) · ωr(A•)

s(E)r(E)

− c(E) · ωr(E)

s(A•)r(A•)

)]≥

0 if µω(A•) = µω(E)n− µω(E)P if µω(A•) < µω(E).

Therefore, for n 0, φn(A•) ≤ φn(E) for all subobjects 0 6= A• → E in A and soE is σn-semistable for all n 0.


Remark D.28. The categories Aβ,ω, with ω, β ∈ NS(X)Q just introduced, are use-ful also in other circumstances. Indeed it has been shown in [154] that they behave“quite naturally” with respect to the category of coherent sheaves. Theorem 0.1in [154] shows that two smooth projective K3 surfaces X and X ′ have equivalentderived categories if and only if there exists β, ω ∈ NS(X)Q, ω ∈ Amp(X) andβ′, ω′ ∈ NS(X ′)Q, ω′ ∈ Amp(X ′) such that Aβ,ω and Aβ′,ω′ are equivalent (asAbelian categories). This fact is then applied to the question of preservation ofstability under a Fourier-Mukai equivalence associated to a locally free universalfamily of µ-stable sheaves ([154, Thm. 0.3]). More precisely, let ω ∈ NS(X) bean ample divisor and assume X ′ is isomorphic to a fine moduli of µω-stable vec-tor bundles on X with Mukai vector v = (r, c, s). Denote the universal family byE• ∈ Db(X ×X ′) and the induced equivalence by Φ = ΦE

•

X→X′ : Db(X)→ Db(X ′).

Then, there exists an ample divisor ω′ ∈ NS(X ′) such that, for any µω-stablevector bundle E on X with µω(E) = −(c · ω)/r, one has either Φ(E) ' C(y)[−2]if [E∨] = y ∈ X ′ or otherwise Φ(E) ' F [−1], for F a µω′ -stable vector bundle onX ′.

These results have been generalized in [296] to obtain a general asymptoticaltheorem on preservation of stability. Unfortunately we cannot summarize herethe complete result. We simply observe that, roughly, [296, Thm. 3.13] yieldsa complete result on preservation of stability under a Fourier-Mukai transform,provided the notion of µ-stability is replaced by that of twisted stability and thedegree is sufficiently large (but universally bounded). The reader should comparethis with the results in Section 4.4 4

D.3.3 The covering map property

In this section we prove that the map π : Stab(X)→ P+0 (X) is a covering on the

preimage of P+0 (X) in Stab(X). This will follow from Bridgeland’s deformation

lemma and will be the key to proving Theorem D.19.

Lemma D.29. Let ‖−‖ be a norm on H1,1(X,Z)C and let f ∈ P+0 (X). Then there

exists a real number r = r(f) > 0 such that

|〈u, v〉| ≤ r‖u‖ · |〈f, v〉|,

for all u ∈ H1,1(X,Z)C and for all v ∈ H1,1(X,Z) ⊗ R with either v2 ≥ 0 orv ∈ ∆(X).

Proof. See [66, Lemma 8.1].

Proposition D.30. The subset P+0 (X) ⊂ H1,1(X,Z)C is open and the restriction

π : π−1(P+0 (X)) ⊂ Stab(X)→ P+

0 (X)


is a covering map (and, in particular, is surjective).

Proof. Fix a norm ‖ − ‖ on NC(X). Take f ∈ P+0 (X) and let r = r(f) be as in

Lemma D.29. For η > 0, define

Dη(f) =

f′ ∈ H1,1(X,Z)C : ‖f′ − f‖ < η/r⊂

openH1,1(X,Z)C.

By Lemma D.29 (and some linear algebra), if η < 1 then Dη(f) ⊂ P+0 (X). Hence

P+0 (X) is open in H1,1(X,Z)C. Now, for all σ ∈ Stab(X) with π(σ) = f, define

Cη(σ) =τ ∈ π−1(Dη(f)) : f(σ, τ) < 1/2

⊂

openStab(X).

By Bridgeland’s deformation lemma, for η > 0 sufficiently small π|Cη(σ) : Cη(σ)→Dη(f) is an homeomorphism. Indeed, for all E• ∈ Db(X) that are σ-stable, aneasy application of the Riemann-Roch theorem shows that v(E•)2 ≥ −2, v(E•) ∈H1,1(X,Z). By Lemma D.29 we have

|〈f′, v(E•)〉 − 〈f, v(E•)〉| ≤ r‖f′ − f‖ · |〈f, v(E•)〉| < η|〈f, v(E•)〉|,

for f′ ∈ Dη(f) and for all E• ∈ Db(X) σ-stable, which is precisely (D.16).

This implies that π−1(P+0 (X)) is the union of full connected components of

Stab(X) (in the sense of Remark D.18(ii)). By using in addition Lemma D.16,π|Cη(σ) is a homeomorphism for any η < sin(πε) if ε < 1/10. As a consequence ofthe uniform choice for η, we have

π−1(Dη(f)) =⊔

σ∈π−1(f)

Cη(σ),

where the union is disjoint by Lemma D.15.

Eventually we need to show that π is surjective on P+0 (X). By Lemma D.25,

π−1(P+0 (X)) is not empty. Indeed, π(σβ,ω) ∈ P+

0 (X), whenever β, ω ∈ NS(X)Q,with ω ample. Set Γ = π(π−1(P+

0 (X))) ⊂ H1,1(X,Z)C. We have just proved thatΓ is open. We need to show that Γ is closed in P+

0 (X). Take a convergent sequencefn → f in P+

0 (X) with fn ∈ Γ, n ∈ N. Consider Dη(f), with η < (1/2) sin(πε).Then there exist N ∈ N and σN ∈ Stab(X) such that π(σN ) = fN ∈ Dη(f).Hence, for all E• ∈ Db(X) σN -stable, we have

|〈f, v(E•)〉 − 〈fN , v(E•)〉| < η|〈f, v(E•)〉| <η

1− η|〈fN , v(E•)〉| < 2η|〈fN , v(E•)〉|.

Again by Lemma D.16 there exists σ ∈ Stab(X) such that π(σ) = f. Therefore Γis a nonempty, open and closed subset of P+

0 (X) and so, since P+0 (X) is connected,

Γ = P+0 (X).


The proof of Proposition D.30 works also for the subset P−0 (defined in thesame way as P+

0 but using the other connected component P−(X) of P(X)).However, there is no known example of stability condition whose central chargetakes values in P−0 (X). A connected component of Stab(X) is called good if itcontains a point σ with π(σ) ∈ P0(X) = P+

0 (X) ∪ P−0 (X). As we saw in theproof of the previous proposition, a good connected component is full. A stabilitycondition will be called good if it lies in a good connected component.

D.3.4 Wall and chamber structure

As an interlude we briefly show that a good connected component of the space ofstability conditions on a K3 surface has a wall and chamber structure for semistableobjects very similar to the one of the ample cone. Let Stab∗(X) be a good con-nected component of Stab(X).

Definition D.31. A set of objects S ⊂ Db(X) has bounded mass with respect toStab∗(X) if

sup mσ(E•) : E• ∈ S <∞

for some (and hence for all) σ ∈ Stab∗(X). 4

An easy consequence of the definition is that the set of Mukai vectors of aset of objects with bounded mass is finite (see [66, Lemma 9.2]).

Proposition D.32. Let S ⊂ Db(X) be a subset with bounded mass and let B ⊂Stab∗(X) be compact. There exists a finite collection Wγγ∈Γ of (not necessar-ily closed) real codimension 1 submanifolds of Stab∗(X) such that any connectedcomponent C ⊂ B \

⋃Wγ has the following property: if E• ∈ S is σ-semistable for

some σ ∈ C, then E• is τ -semistable for all τ ∈ C. Moreover, if v(E•) is primitive,then E• is τ -stable, for all τ ∈ C.

Proof. We only show how to construct the walls. Define T as the set of nonzeroobjects G• in Db(X) for which there exist σ ∈ B and E• ∈ S such that mσ(G•) ≤mσ(E•). For example, if G• is a σ-semistable HN factor of E• for some σ ∈ B, thenG• ∈ T . Since B is compact, T has bounded mass. Then the set of Mukai vectorsof T is finite. Let us denote it by vii∈I . Set

Γ = (i, j) ∈ I × I : vi 6= αvj , for all α ∈ R .

Finally, for γ = (i, j) ∈ Γ, define

Wγ = σ = (Z,P) ∈ Stab∗(X) : Z(vi)/Z(vj) ∈ R>0 .

For the conclusion of the proof, see [66, Prop. 9.3].


Proposition D.33. Let S ⊂ Db(X) be a subset with bounded mass and assume thatfor all E ∈ S, v(E) is primitive. Then the subset

σ ∈ Stab∗(X) : all E• ∈ S are σ-stable ⊂ Stab∗(X)

is open.

Proof. See [66, Prop. 9.4].

D.3.5 Sketch of the proof of Theorem D.19

Define a subset U(X) ⊂ Stab(X) by

U(X) = σ ∈ Stab∗(X) : σ is good and

Oxx∈X are all σ-stable of the same phase .

We now list, mostly without proof, some important properties of the subset U(X).

(i) By Proposition D.33, U(X) is open. By Example D.26, it is nonempty.

(ii) By [66, Prop. 10.3], if σ ∈ U(X) there exists a unique element G ∈Gl

+

2 (R) and there exist β, ω ∈ NS(X)R, ω ∈ Amp(X) such that σ · G =

(Zβ,ω,Aβ,ω). In particular, the action of Gl+

2 (R) is free on U(X). A sectionis given by

V (X) =σ ∈ U(X) :

π(σ) = exp(β + iω) ∈ H1,1(X,Z)C, ω ∈ Amp(X)φσ(Ox) = 1 for all x ∈ X

.

(iii) By [66, Prop. 11.2], the map π : V (X)→ L(X), where

L(X) = exp(β + iω)

with β, ω ∈ H1,1(X,Z)R and ω ∈ Amp(X) as in Lemma D.24, is a homeo-morphism. In particular, all stability conditions arising from the construc-tion of Lemma D.24 admit HN-filtrations and are locally finite. Moreover,an easy check shows that π(U(X)) ⊂ P+

0 (X) and that σ ∈ U(X) is uniquelydetermined by π(σ) up to even shifts.

By (iii), U(X) is connected. Let Stab†(X) be the connected component ofStab(X) which contains U(X). By definition, it is a good connected component.The important fact about U(X) is that a “general” point of its boundary inStab†(X) can be explicitly described (see [66, Sec. 12,13]). From this, we candeduce the following additional two facts:


(iv) For all τ ∈ Stab†(X), there exists Φ ∈ Aut(Db(X)) such that fΦ((0, 0, 1)) =(0, 0, 1) in H1,1(X,Z) and Φ(τ) ∈ U(X), where U(X) denotes the closure ofU(X) in Stab†(X).

(v) π(Stab†(X)) = P+0 (X).

We can now complete the proof of Theorem D.19.

Sketch of the proof of Theorem D.19. The first part of the claim follows from Propo-sition D.30 and (v) above. Moreover, clearly Aut†0(Db(X)) preserves π. Hence weonly have to show that, if σ, τ ∈ Stab†(X) are such that π(σ) = π(τ), there exists aunique Φ ∈ Aut+

0 (Db(X)) such that Φ(σ) = τ . Since π is a covering, it is sufficientto prove this for σ = σβ,ω as in Section D.3.2.

Uniqueness: assume Φ(σ) = σ. Then, since Ox is σ-stable of phase 1, for allx ∈ X, Φ(Ox) is σ-stable of phase 1, too. But an easy check shows that the onlystable objects in σ of phase 1 and Mukai vector (0, 0, 1) are the skyscraper sheaves.Hence, for all x ∈ X, there exists y ∈ X such that Φ(Ox) ' Oy. By Corollary1.12, Φ ' (L ⊗ −) f∗, for f ∈ Aut(X) an automorphism of X and L ∈ Pic(X).Since, by assumption, fΦ = Id, then L ' OX . But then, by the Torelli theoremfor K3 surfaces (Theorem 4.10), Φ = Id, as wanted.

Existence: assume that τ ∈ Stab†(X) satisfies π(τ) = π(σ). By (iv), thereexists Φ ∈ Aut(Db(X)) such that fΦ((0, 0, 1)) = (0, 0, 1) in H1,1(X,Z) andΦ(τ) ∈ U(X). By suitably modifying σ we can assume that both σ and Φ(τ)are in U(X). Now, since fΦ is an isometry with respect to the Mukai pairing,it can be easily checked that fΦ((1, 0, 0)) = exp(c1(L)), for some L ∈ Pic(X).Hence, by composing Φ with the functor L ⊗ − (which preserves U(X)), we canassume fΦ((1, 0, 0)) = (1, 0, 0). Therefore, the action of Φ on H(X,Z) preservesthe grading H0(X,Z)⊕H2(X,Z)⊕H4(X,Z). Since σ and Φ(τ) are in U(X), theinduced Hodge isometry on H2(X,Z) is effective, i.e., it preserves the Kahler coneof X. Again, by the Torelli theorem, there exists an automorphism f ∈ Aut(X)such that f∗ = fΦ on H2(X,Z). By composing with f∗, we can eventually assumefΦ = Id. But so π(σ) = π(Φ(τ)). By (iii), σ = Φ(τ)[2k], for some integer k ∈ Z.Since the shift by 2 functor induces the identity in cohomology, we are done.

We conclude this section by presenting, without proof, an example (which isa particular case of a more general construction given by Meinhardt and Partschin [217]) of a space of stability conditions which is not connected.

Example D.34. Let X be a smooth projective K3 surface over C. Consider thederived category Db

(1)(X) defined as the Verdier quotient of Db(X) by its thicksubcategory consisting of complexes whose cohomologies are supported in codi-mension greater than 1. In [217, Thm. 1.2] it is shown that Db

(1)(X) is equivalentto the bounded derived category of Coh(1)(X), which is an Abelian category of di-mension 1 obtained as a quotient of Coh(X) by sheaves supported in codimension

D.4. Moduli stacks and invariants of semistable objects on K3 surfaces 385

greater than 1. Let Stab(X(1)) be the space of locally finite numerical stabilityconditions on Db

(1)(X). Then, by [217, Thm. 1.3], Stab(X(1)) is isomorphic to a

disjoint union of free Gl+

2 (R)-orbits

Stab(X(1)) '⊔

ω∈C(X)/R>0

σω · Gl+

2 (R),

where

C(X) = ω ∈ NS(X)R : infω ·D : D ⊂ X effective divisor on X > 0 ,

R>0 acts on C(X) by multiplication, and σω has heart Coh(1)(X) and stabilityfunction Zω(E) = −ω · c(E) + ir(E), for v(E) = (r(E), c(E), s(E)), E ∈ Coh(X).

4

D.4 Moduli stacks and invariants of semistable objects

on K3 surfaces

In this section we examine a further problem related to stability conditions: theconstruction of moduli spaces of stable objects. This will require some notions andtechniques, such as those of stacks, 2-categories and 2-functors, that we have notused in this book. For quite readable introductions to these topics, the reader mayrefer to [193, 126].

We concentrate on the case of K3 surfaces described in the previous sec-tion, where the situation is reasonably well understood, thanks to the beautifulpaper [281] (but see also [7], where some specific moduli spaces are studied inmore detail). In the final part we also sketch the problem of studying invariantsby counting semistable objects. The main results about K3 surfaces presentedhere were conjectured by D. Joyce in [169] and have been proved by Y. Toda in[281]. Some computations of such invariants are contained in [218]. For very re-cent developments about the very interesting subject of counting invariants andwall-crossing formulas see, for example, [169, 167, 72, 244, 245, 280, 31, 279, 172].

D.4.1 Moduli stack of semistable objects

Let X be a smooth projective K3 surface over C. Let SchC be the site of locallyNoetherian schemes over C (endowed with the etale topology). Define a 2-functor,with values in the category Grp of groupoids, M : SchC → Grp, by mapping a C-scheme S to the groupoid M(S) whose objects consist of those E• ∈ DS-perf (X×S)which satisfy Exti(E•s, E•s) = 0, for all i < 0 and all s ∈ S. Here

DS-perf (X × S) ⊂ D(OX×S-Mod)


denotes the derived category of S-perfect complexes (an S-perfect complex is acomplex of OX×S-modules which, locally over S, is quasi-isomorphic to a boundedcomplex of coherent sheaves which are flat over S) and E•s is the derived restrictionof E• to X × s → X × S. The main theorem in [197] (generalizing results in[160]) shows that M is an Artin stack, locally of finite type over C.

Fix σ = (Z,P) ∈ Stab†(X), φ ∈ R, and v ∈ H1,1(X,Z). We define a sub2-functor M(v,φ)(σ) ⊂ M by considering the objects E• ∈ DS-perf (X × S) whoserestrictions E•s belong to P(φ) and have Mukai vector v for all s ∈ S. The mainresult of this section, [281, Thm. 1.4], is a first step in understanding the firstquestion at page 368.

Theorem D.35. The 2-functor M(v,φ)(σ) is an Artin stack of finite type over C.

We shall presently describe a proof of this theorem.

Example D.36. Take σ ∈ U(X), v = (0, 0, 1), and φ = 1. By what we haveseen in Section D.3, the only semistable objects verifying these conditions are theskyscraper sheaves, which are σ-stable and have as automorphism group the torusGm ' C∗. Hence

M((0,0,1),1)(σ) ' [X/Gm] ,

where Gm acts trivially on X. 4Example D.37. Let ω ∈ NS(X) be an ample divisor. Take v = (r, c, s) ∈ H1,1(X,Z)with r > 0, c · ω > 0 and gcd(exp〈nω), v〉 : n ∈ Z = 1. Proposition 6.4 andLemma 6.5 in [281] show the existence of a universal bound for the situationdescribed in Example D.27. Namely, there exists a positive integer N such thatthe semistable objects in σN = σ0,Nω that are in A0,ω with Mukai vector v areprecisely the ω-Gieseker stable (torsion-free) sheaves on X with Mukai vectorv. Denote by Mv(ω) their fine moduli space. Notice that Mv(ω) is a smooth,projective, symplectic variety. We have

M(v,φ)(σN ) ' [Mv(ω)/Gm] ,

where again Gm acts trivially on Mv(ω) and φ ∈ (0, 1] is such that

Z0,Nω(v)/|Z0,Nω(v)| = exp(iπφ) .

4

D.4.2 Sketch of the proof of Theorem D.35

We divide the proof in a few steps. Fix σ = (Z,P) ∈ Stab†(X), φ ∈ R, andv ∈ H1,1(X,Z). Denote by M (v,φ)(σ) the subset of objects in Db(X) consisting ofthe semistable objects in σ of phase φ and Mukai vector v.


Step 1. (Understanding the problem.) Let us recall the definition of boundednessfor a set of objects in Db(X). Let S ⊂ Db(X) be a set of objects in Db(X).We say that S is bounded if there exists a C-scheme Q of finite type and F• ∈DQ-perf (X ×Q) such that any object E• ∈ S is isomorphic to F•q for some closedpoint q ∈ Q.

Lemma D.38. Assume that the following two conditions hold:

(i) M(v,φ)(σ) ⊂M is an open substack.

(ii) M (v,φ)(σ) ⊂ Db(X) is bounded.

Then M(v,φ)(σ) is an Artin stack of finite type over C.

Proof. Let M → M be an atlas of M. The openness of M(v,φ)(σ) implies thatthere is an open subset M0 ⊂ M which gives a surjective smooth morphismM0 →M(v,φ)(σ). Hence M(v,φ)(σ) is an Artin stack, locally of finite type over C.

Moreover, the boundedness of M (v,φ)(σ) yields a surjection Q→M0, whereQ is a scheme of finite type over C. But then M0 is of finite type, too, and thenM(v,φ)(σ) becomes an Artin stack of finite type over C.

Hence, by Lemma D.38, to prove Theorem D.35 we only need to show thatconditions (i) and (ii) in the statement hold. The first step is to give a sufficientcondition for (i).

Lemma D.39. Assume that the following is true:

(i’) if, for a smooth quasi-projective variety S and E ∈M(S), the locus

S0 = s ∈ S : v(E•s) = v and E•s ∈ P(φ) (D.5)

is nonempty, then it contains a nonempty open subset of S.

Then condition (i) above holds.

Proof. Since M is an Artin stack of locally finite type over C, it is sufficient toprove that, for any affine scheme S of finite type over C and for any E ∈ M(S),the locus S0, defined analogously as in (D.5), is open in S.

Assume such S0 is nonempty. By using resolution of singularities and (a′), itis not difficult to see that there exists a nonempty open subset U1 ⊂ S such thatU1 ⊂ S0. Let Z1 = S \U1. If Z1 ∩ S0 is empty, we have S0 = U1 and we are done.Assume Z1 ∩ S0 6= ∅. Take the pull-back EZ1 ∈ M(Z1) of E to Z1 and apply thesame argument. Then we obtain an open subset U2 ⊂ Z1 such that U2 ⊂ (Z1∩S0)


and a closed subset Z2 = Z1 \ U2. Repeating the same argument again, we get asequence of closed subsets in S

. . . ⊂ Zn ⊂ Zn−1 ⊂ . . . ⊂ Z1

which must terminate since S is Noetherian. Then Z = ∩iZi is a closed subset ofS and we have S0 = S \ Z, which is open as wanted.

Step 2. (Reduction of (i’) to generic flatness for algebraic stability conditionswhich verify (ii).) A stability condition σ ∈ Stab(X) is called algebraic if the imageof its central charge is contained in Q + iQ ⊂ C. For example, if β, ω ∈ NS(X)Q,with ω ample, then σβ,ω (when it exists, e.g., under the assumption of LemmaD.24) is algebraic. Note that the heart of an algebraic stability condition is aNoetherian Abelian category by [1, Thm. 5.0.1].

Let S be a smooth projective variety over C and let L be an ample linebundle (for generalizations to arbitrary schemes, but giving up some results thatwill be important for us, see [253]).

The authors of [1], given any Noetherian heart A of a bounded t-structureon Db(X), construct a heart of a bounded t-structure on Db(X × S) which isNoetherian and independent of the choice of L:

AS =F ∈ Db(X × S) : (pX)∗(F ⊗ p∗S(Ln)) ∈ A, for all n 0

,

where pS and pX denote the two projections from X×S to S and X, respectively.Moreover, if S is smooth and quasi-projective, we can define AS as the essentialimage of AS under the functor (Id × j)∗ : Db(X × S) → Db(X × S), where S isa smooth compactification of S and j : S → S denotes the natural inclusion. Itcan be proved (see [1], Theorem 2.7.2 and Lemma 3.2.1) that AS is the heart ofa bounded t-structure and that its definition is independent of the chosen smoothcompactification.

Lemma D.40. Let σ = (Z,P) ∈ Stab†(X) be an algebraic stability condition suchthat M (v,φ)(σ) is nonempty. Assume that generic flatness holds for Aφ = P((φ−1, φ]), that is, that for a smooth quasi-projective variety S and E• ∈ AφS, there isan open subset U ⊂ S such that, for each s ∈ U , E•s ∈ Aφ. Then condition (i’)holds.

Proof. Note that since σ algebraic and M (v,φ)(σ) is nonempty, Aφ is a NoetherianAbelian category. In particular, it makes sense to consider AφS for a smooth quasi-projective variety S.

Let E• ∈ M(S) and assume the locus S0 in (D.5) nonempty. Take s ∈ S0.Then E•s ∈ P(φ) ∈ Aφ. By the open heart property (see [1, Thm. 3.3.2]), there


exists an open neighborhood s ∈ U ⊂ S such that E•U ∈ AφU . By applying thegeneric flatness condition to U , we have the existence of an open subset U ′ ⊂U ⊂ S such that for each s′ ∈ U ′, E•s′ ∈ Aφ. Since v(E•s′) = v(E•s), we haveZ(E•s′)/|Z(E•s′)| = exp(iπφ). But then E•s′ is semistable, i.e., E•s′ ∈ P(φ). Thisshows that U ′ ⊂ S0.

We now want to reduce the condition in this lemma to generic flatness forP((0, 1]). This is the main technical result in [281].

Lemma D.41. Let σ = (Z,P) ∈ Stab†(X) be an algebraic stability condition suchthat M (v,φ)(σ) is nonempty. Assume the following two conditions are satisfied:

(a) Generic flatness holds for A = P((0, 1]), i.e., for a smooth quasi-projectivevariety S and E ∈ AS, there is an open subset U ⊂ S such that, for eachs ∈ U , Es ∈ A.

(b) M (v,φ)(σ) ⊂ Db(X) is bounded.

Then generic flatness holds for Aφ = P((φ− 1, φ]).

Proof. See [281], Proposition 3.18.

Hence, in particular, if (a) and (b) hold for an algebraic stability conditionσ, then M(v,φ)(σ) is an Artin stack of finite type over C.

Step 3. Here we state, without proving, the main reduction step:

Theorem D.42. Assume (a) and (b) hold for all stability conditions in Stab†(X)of the form σβ,ω with β, ω ∈ NS(X)Q. Then M(v,φ)(σ) is an Artin stack of finitetype over C, for all σ ∈ Stab†(X), φ ∈ R and v ∈ H1,1(X,Z).

Proof. See[281], Theorem 1.3

At this point, to complete the proof of Theorem D.35 we only need to showthat boundedness and generic flatness for Aβ,ω hold for σβ,ω, β and ω rational.

One should note that the previous theorem is stated in [281] in a greatergenerality, namely, for any smooth projective varieties X; the set of stability con-ditions of the form σβ,ω with β and ω rational is replaced by a subset of algebraicstability conditions which is dense in a fundamental domain for the action of theautoequivalence group. However a further assumption on subsets of bounded massmust be added (see [281], Assumption 3.1).

Step 4. (Proof of generic flatness for σβ,ω, with β, ω ∈ NS(X)Q.) We first sketchthe proof of generic flatness for A = Aβ,ω, β, ω ∈ NS(X)Q, ω ∈ Amp(X).


Lemma D.43. For a smooth quasi-projective variety S and E• ∈ AS, there is anopen subset U ⊂ S such that E•s ∈ A for all s ∈ U .

Proof. Clearly we can assume that S is projective. Pick an ample line bundleL ∈ Pic(X) and assume E• ∈ AS . By definition, we have

(pX)∗(E• ⊗ p∗S(Ln)) ∈ A,

for all n 0. By using the spectral sequence

Ep,q2 = RipX∗(Hq(E•)⊗ p∗S(Ln)) =⇒ Rp+qpX∗(E• ⊗ p∗S(Ln)),

the fact that pX∗(E ⊗ p∗S(Ln)) has only nonzero cohomologies in degree −1 and 0implies that the same holds true for E•.

The existence of relative Harder-Narasimhan filtrations (see, e.g.,Thm. 2.3.2in [155]) gives an open subset U ⊂ S and filtrations by coherent sheaves

0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk−1 ⊂ Fk = H−1(E•)U ,

0 = T 0 ⊂ T 1 ⊂ . . . ⊂ T l−1 ⊂ T l = H0(E•)U(D.6)

such that all F i and T i are flat sheaves on U . Moreover, for all s ∈ U , thefiltrations in (D.6) give the Harder-Narasimhan filtrations (with respect to ω-Gieseker stability) of H−1(E•)s and of H0(E•)s, respectively.

Now, using the definition of Aβ,ω and Proposition 3.5.3 in [1] (which roughlysays that (a) holds for a dense subset in S), it is easy to see that E•s ∈ Aβ,ω forall s ∈ U .

Step 5. (Proof of (b) for σβ,ω, with β, ω ∈ NS(X)Q.) Finally, we are reduced toshow boundedness for σβ,ω ∈ Stab†(X), with β, ω ∈ NS(X)Q.

Lemma D.44. The subset M (v,φ)(σβ,ω) ⊂ Db(X) is bounded.

Proof. We only give the basic ideas of the main argument. For the details werefer to Section 4.5 of [281]. First of all, by shifting, it is sufficient to show theboundedness of

S = E• ∈ Aβ,ω : v(E•) = v and E• semistable in σβ,ω .

Consider the following three sets of objects:

T ′ =H0(E•)tor : E• ∈ S

,

T =H0(E•)/H0(E•)tor : E• ∈ S

,

F =H1(E•) : E• ∈ S

.


Clearly it is sufficient to show that each of the previous subsets is bounded. Thisis achieved by showing that the sets of Mukai vectors of possible µω-semistablefactors of every object in either T or F are finite and similarly that the same istrue for (β, ω)-twisted semistable factors of every object in T ′ (for the notion oftwisted stability, see [213]). Hence the boundedness of T ′, T , and F follows fromthe corresponding one for the usual notions of stability for sheaves.

By Theorem D.42, this completes the proof of Theorem D.35.

D.4.3 Counting invariants and Joyce’s conjecture for K3 surfaces

In this section we try to understand the second question at page 368. First webriefly recall the work of Joyce in [169].

Let K(VarC) be the Grothendieck ring of quasi-projective varieties over C,i.e., the Z-module generated by the isomorphism classes of quasi-projective vari-eties with relations generated by X − Y − (X \ Y ) for closed subschemes Y ⊂ X.The product is induced by the formula X · X ′ = X × X ′. Suppose Λ is a com-mutative associative Q-algebra (with identity 1) and γ : K(Var) → Λ a motivicinvariant of varieties, i.e., a ring homomorphism. Set L = γ(A1

C). We assume thatboth L and Lk−1 are invertible, for k ≥ 1. An example is given by Λ = Q(z) withγ given by the virtual Poincare polynomial of X (for X smooth and projectivethis is nothing but the usual Poincare polynomial).

Let X be a smooth and projective K3 surface over C. Fix ω ∈ NS(X) anample divisor and v ∈ H1,1(X,Z) and fix a motivic invariant of varieties γ. In [169,Sect. 6] is constructed a (weighted) system of invariants Jv(ω) ∈ Λ “counting” ω-Gieseker semistable sheaves E ∈ Coh(X) with Mukai vector v(E) = v. For example,if we are in the situation as in Example D.37, then Jv(ω) = γ(Mv(ω)). In [169,Thm. 6.24] it is shown that Jv(ω) does not depend on ω. Denote it by Jv.

Now the natural question [169, Conj. 6.25] is whether Joyce’s theorem gen-eralizes to Bridgeland’s stability conditions. The answer is yes:

Theorem D.45. (Toda) Fix a motivic invariant γ : K(Var)→ Λ for some commu-tative associative Q-algebra Λ. For σ ∈ Stab†(X) and v ∈ H1,1(X,Z), there existsa weighted system of invariants Jv(σ) ∈ Λ “counting” semistable objects in σ withMukai vector v, such that

(i) Jv(σ) does not depend on the choice of σ.

(ii) If v ∈ C(X) = im(Coh(X) v−→ H1,1(X,Z)

), then Jv(σ) = Jv(σ).

In the next section we shall sketch how to construct the invariants Jv(σ) andprove Theorem D.45. Here we only make a few comments. Denote by Aut†(Db(X))


the subgroup of the autoequivalence group of Db(X) consisting of the autoequiv-alences which preserve the connected component Stab†(X). Then, by TheoremD.45, for all Φ ∈ Aut†(Db(X)), we have

Jv(σ) = JfΦ(v)(Φ(σ)) = Jf

Φ(v)

for some (and then any) σ ∈ Stab†(X). In particular this may be useful forconstructing some interesting automorphic functions on Stab†(X), i.e., functionswhich are invariant under autoequivalences. An example, as pointed out in [169],is provided by the map fk : Stab†(X) → Λ ⊗Z C (k ∈ Z) defined by (ignoringconvergence problems)

σ = (Z,P) 7→∑

v∈H1,1(X,Z)\0

Jv(σ)Z(v)k

.

As pointed out in the Introduction to this appendix, it seems quite interesting tounderstand how this generalizes to higher dimensions. More precisely, one is inter-ested in defining invariants for Calabi-Yau threefolds arising from moduli spacesof “stable” objects in Db(X) (see Conjecture 6.30 and Section 7 in [169]). Herethe word “invariant” refers to invariance with respect to deformations (i.e., theseobjects should behave like generalized Donaldson-Thomas invariants) and withrespect to the change of stability conditions (i.e., there should exist wall-crossingformulas which explain how they vary under a change of the stability condition).At the moment these constructions turn out to be quite problematic. Most suchproblems are described in detail in Sections 6 and 7 of [169]. Here we just pointto some recent literature. For examples, explanations, and relevant conjecturesthe reader is referred to [244, 245]. For the “stability condition” interpretation ofthese results see [31, 279] (where a different notion of stability condition on de-rived categories is introduced: a sort of limit of the notion of Bridgeland stabilitycondition as presented in this note). For an attempt in understanding Joyce’s workvia certain geometric structures on the space of stability conditions (but only forAbelian categories and under certain finiteness conditions) see [167, 72]. A moregeneral approach which also covers derived categories of Calabi-Yau threefolds isgiven in [172].

D.4.4 Some ideas from the proof of Theorem D.45

For the construction of the invariants Jv(σ) we need to extend a motivic invariantof varieties to stacks. Here we shall be quite sketchy; the reader is referred to[168, 281] for full details. Consider the ring SF(StC) defined as the Q-vector spacesgenerated by the isomorphism classes of Artin stack of finite type over C with affinestabilizers modulo the ideal generated by the relations X −Y − (X \Y) for closed


substacks Y ⊂ X . The product is induced by the Cartesian product of stacks overC. An algebraic C-group G will be called special if every principal G-bundle islocally trivial.

In Theorem 4.9 in [168] it is shown that, given a motivic invariant of varietiesγ : K(VarC)→ Λ, there exists a unique morphism of Q-algebras γ′ : SF(StC)→ Λsuch that, if G is a special group (and so γ(G) is invertible in Λ) acting on aquasi-projective variety X, then γ′([X/G]) = γ(X)/γ(G).

Take now v ∈ H1,1(X,Z), φ ∈ R, and σ = (Z,P) ∈ Stab†(X). By TheoremD.35, M(v,φ)(σ) is an Artin stack of finite type over C.

Definition D.46. Given a motivic invariant γ, define Iv(σ) ∈ Λ as follows:

Iv(σ) =

γ(M(v,φ)(σ)), if Z(v) 6= 0 and Z(v)/|Z(v)| = exp(iπφ),0, if Z(v) = 0.

4

Notice that, in the previous definition, Iv(σ) is independent of the choice ofφ. To introduce the invariants Jv(σ) we still need to introduce some definitions.Set Cσ(φ) = im(P(φ)→ H1,1(X,Z)) \ 0.

Definition D.47. Let Cvv∈H1,1(X,Z) be a set of formal variables parameterizedby H1,1(X,Z). Define:

(i) a ring H =⊕

v∈H1,1(X,Z)

Λ · Cv with product ∗ induced by

Cv ∗ Cv′ = L〈v,v′〉 · Cv+v′ ,

for all v, v′ ∈ H1,1(X,Z), where L = γ(A1C).

(ii) δv(σ) = Iv(σ) · Cv.

(iii)

εv(σ) =

∑

v1+...+vmvi∈Cσ(φ)

(−1)m−1

m δv1(σ) ∗ . . . ∗ δvm(σ), if Z(v)/|Z(v)| = exp(iπφ),

0, otherwise.

4

It is not too difficult to check that the sum in the definition of εv(σ) is finite(see Lemma 5.12 in [281]). Then εv(σ) = B ·Cv, for some B ∈ Λ. Define Jv(σ) ∈ Λ


by Jv(σ) = B · (L− 1). More explicitly, we have

Jv(σ) =∑

v1+...+vmvi∈Cσ(φ)

(−1)m−1(L− 1)m

LPj>i〈vj ,vi〉 m∏

Ivi

i=1

(σ).

The definition of Jv(ω) in [169, Def. 6.22] is analogous (replacing M(v,φ)(σ)with the stack Mv(ω) of ω-Gieseker semistable sheaves of Mukai vector v andthe condition vi ∈ Cσ(φ) with the condition that the Hilbert polynomials are thesame).

To prove Theorem D.45, we first need to check that the previous definitionof Jv(σ) is indeed independent from σ. Take σ and τ in Stab†(X). Choose a pathα : [0, 1] → Stab†(X) such that α(0) = σ and α(1) = τ . Consider a open subsetB0 ⊂ Stab†(X) such that α([0, 1]) ⊂ B0 and its closure B is compact. Define asubset S ⊂ Db(X) by

S =E• ∈ Db(X) there exists σ′ = (Z ′,P ′) ∈ B such that

E• is σ-semistable with |Z ′(E•)| ≤ |Z ′(v)| .

Since B is compact, then S has bounded mass. By Proposition D.32, there existsa wall and chamber structure Wl on B with respect to S.

We may assume that the set of points K ⊂ [0, 1] on which σt = (Zt,Pt) =α(t) is algebraic and Pt((ψ − 1, ψ]) satisfies generic flatness for any ψ such thatP(ψ) is nonempty, is dense in [0, 1]. Take s0, s1, . . . , sN+1 ∈ [0, 1] and t±i ∈(si, si+1) ∩K such that

• For 1 ≤ i ≤ N , si ∈Wl for some Wl and s0 = 0, sN+1 = 1.

• For any t ∈ (si, si+1), we have α(t) /∈Wl, for all l.

• σt+i ∈ Bε(σsi+1), σt−i ∈ Bε(σsi), with ε > 0 fixed sufficiently small.

Hence we only need to show the following two cases:

(i) σ and τ are in the same chamber.

(ii) σ is contained in a chamber and τ is in a boundary of that chamber.

Case (i) is proved in [281, Prop. 5.17]. The basic idea is that an object in Sis semistable in σ if and only if it is semistable in τ .

Case (ii) is proved in [281, Prop. 5.23], by translating a combinatorial ar-gument of Joyce in [169] to the algebra H previously defined, which shows thatεv(σ) = εv(τ) if σ and τ are sufficiently close (which we may assume by case (i)).

For v ∈ H1,1(X,Z), define Jv = Jv(σ) for some (any) stability conditionσ ∈ Stab†(X). To conclude the proof of Theorem D.45 we only need to show that


if v ∈ C(X), then Jv = Jv. This is proved in [281, Sect. 6]. The main point isessentially a generalization of Example D.37. Indeed, if v ∈ C(X), then it is easyto see that, up to tensoring by a line bundle (operation which does not changeJv, i.e., for a line bundle L ∈ Pic(X), Jv = Jv·ch(L) and Jv = Jv·ch(L)), we canreduce to the case where v = (r, c, s) with either ω · c > 0 (ω an ample divisor)or r = c = 0. Then to prove the theorem, it is enough to compare Jv(σ0,kω) andJv(ω), for k 0.

This is done in Proposition 6.4 and Lemma 6.5 of [281], by showing firstthat, in the above situation, if φk ∈ (0, 1] is such that Z0,kω(v)/|Z0,kω(v)| =exp(iπφk), then there exists an integer N > 0 such that for all k ≥ N and allv′ ∈ Cσ0,kω (φk) with |=(Z0,ω(v′))| ≤ |=(Z0,ω(v))|, any E ∈M (v′,φk)(σ0,kω) is a ω-Gieseker semistable sheaf. Then, vice versa, if v′ has the same Hilbert polynomialas v and |=(Z0,ω(v′))| ≤ |=(Z0,ω(v))|, any ω-Gieseker semistable sheaf of Mukaivector v′ is σ0,kω-semistable. A short technical computation yields the desiredequality between Jv(σ0,kω) and Jv(ω).

References

[1] D. Abramovich and A. Polishchuk, Sheaves of t-structures and valua-tive criteria for stable complexes, J. reine angew. Math., 590 (2006), pp. 89–130.

[2] L. Alonso Tarrıo, A. Jeremıas Lopez, and J. Lipman, Studies in du-ality on Noetherian formal schemes and non-Noetherian ordinary schemes,Contemporary Mathematics, vol. 244, American Mathematical Society,Providence, RI, 1999.

[3] A. B. Altman and S. L. Kleiman, Compactifying the Picard scheme,Adv. in Math., 35 (1980), pp. 50–112.

[4] B. Andreas and D. Hernandez Ruiperez, Comments on N = 1 het-erotic string vacua, Adv. Theor. Math. Phys., 7 (2003), pp. 751–786.

[5] , Fourier-Mukai transforms and applications to string theory, RACSAMRev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat., 99 (2005), pp. 29–77.

[6] B. Andreas, S.-T. Yau, G. Curio, and D. Hernandez Ruiperez,Fibrewise T -duality for D-branes on elliptic Calabi-Yau, J. High EnergyPhys., 3 (2001), pp. 13, paper 20 (electronic).

[7] D. Arcara and A. Bertram, Bridgeland-stable moduli spaces for K-trivialsurfaces. With an appendix by M. Lieblich. Preprint arXiv:0708.2247.

[8] D. Arinkin and A. Polishchuk, Fukaya category and Fourier trans-form, in Winter School on Mirror Symmetry, Vector Bundles and LagrangianSubmanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., vol. 23,Amer. Math. Soc., Providence, RI, 2001, pp. 261–274.

[9] P. S. Aspinwall, D-branes, Π-stability and θ-stability, in Snowbird lectureson string geometry, Contemp. Math., vol. 401, Amer. Math. Soc., Provi-dence, RI, 2006, pp. 1–13.

398 References

[10] P. S. Aspinwall and R. Y. Donagi, The heterotic string, the tangentbundle and derived categories, Adv. Theor. Math. Phys., 2 (1998), pp. 1041–1074.

[11] P. S. Aspinwall and J. Louis, On the ubiquity of K3 fibrations in stringduality, Phys. Lett. B, 369 (1996), pp. 233–242.

[12] P. S. Aspinwall and I. V. Melnikov, D-branes on vanishing del Pezzosurfaces, J. High Energy Phys., 12 (2004), pp. 30, paper 42 (electronic).

[13] A. Astashkevich, N. Nekrasov, and A. Schwarz, On noncommutativeNahm transform, Comm. Math. Phys., 211 (2000), pp. 167–182.

[14] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer.Math. Soc., 85 (1957), pp. 181–207.

[15] , Vector bundles over an elliptic curve, Proc. London Math. Soc. (3),7 (1957), pp. 414–452.

[16] , Geometry of Yang-Mills fields, Scuola Normale Superiore Pisa, Pisa,1979.

[17] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362(1978), pp. 425–461.

[18] M. F. Atiyah and I. M. Singer, The index of elliptic operators. IV, Ann.of Math. (2), 93 (1971), pp. 119–138.

[19] , Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci.U.S.A., 81 (1984), pp. 2597–2600.

[20] M. F. Atiyah and R. S. Ward, Instantons and algebraic geometry, Comm.Math. Phys., 55 (1977), pp. 117–124.

[21] A. C. Avram, M. Kreuzer, M. Mandelberg, and H. Skarke, Search-ing for K3 fibrations, Nuclear Phys. B, 494 (1997), pp. 567–589.

[22] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Com-pact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete(3), vol. 4, Springer-Verlag, Berlin, second enlarged ed., 2004.

[23] C. Bartocci and I. Biswas, Higgs bundles and the Fourier-Mukai trans-form, Southeast. Asian Bull. Math., 25 (2001), pp. 201–207.

[24] C. Bartocci, U. Bruzzo, and D. Hernandez Ruiperez, A Fourier-Mukai transform for stable bundles on K3 surfaces, J. reine angew. Math.,486 (1997), pp. 1–16.

References 399

[25] , Moduli of reflexive K3 surfaces, in Complex analysis and geometry(Trento, 1995), Pitman Res. Notes Math. Ser., vol. 366, Longman, Harlow,1997, pp. 60–68.

[26] , Existence of µ-stable vector bundles on K3 surfaces and the Fourier-Mukai transform, in Algebraic geometry (Catania, 1993/Barcelona, 1994),Lecture Notes in Pure and Appl. Math., vol. 200, Dekker, 1998, pp. 245–257.

[27] , A hyper-Kahler Fourier transform, Differential Geom. Appl., 8 (1998),pp. 239–249.

[28] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, and J. M.

Munoz Porras, Mirror symmetry on K3 surfaces via Fourier-Mukai trans-form, Comm. Math. Phys., 195 (1998), pp. 79–93.

[29] , Relatively stable bundles over elliptic fibrations, Math. Nachr., 238(2002), pp. 23–36.

[30] C. Bartocci and M. Jardim, A Nahm transform for instantons overALE spaces, in Clifford Algebras: Applications in Mathematics, Physics andEngineering, Progress in Mathematical Physics, vol. 34, Birkhauser, Boston,MA, 2003, pp. 155–166.

[31] A. Bayer, Polynomial Bridgeland stability conditions and the large volumelimit. Preprint arXiv:0712.1083.

[32] A. Beauville, Quelques remarques sur la transformation de Fourierdans l’anneau de Chow d’une variete abelienne, in Algebraic geometry(Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin,1983, pp. 238–260.

[33] A. A. Beılinson, Coherent sheaves on Pn and problems in linear algebra,Funktsional. Anal. i Prilozhen., 12 (1978), pp. 68–69. English translation:Functional Anal. Appl. 12 (1978), no. 3, 214–216 (1979).

[34] , The derived category of coherent sheaves on Pn, Selecta Math. Soviet.,3 (1983/84), pp. 233–237.

[35] A. A. Beılinson, J. Bernstein, and P. Deligne, Faisceaux pervers, inAnalysis and topology on singular spaces, I (Luminy, 1981), Asterisque, vol.100, Soc. Math. France, Paris, 1982, pp. 5–171.

[36] A. A. Beılinson and A. Polishchuk, Torelli theorem via Fourier-Mukaitransform, in Moduli of abelian varieties (Texel Island, 1999), Progress inMathematics, vol. 195, Birkhauser, Basel, 2001, pp. 127–132.

400 References

[37] O. Ben-Bassat, Twisting derived equivalences. Preprint arXiv:math/0606631.

[38] A. Bergman, Stability conditions and branes at singularities. PreprintarXiv:hep-th/0702092.

[39] J. Bernstein and V. Lunts, Equivariant sheaves and functors, LectureNotes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994.

[40] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrerGrenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987.

[41] O. Biquard and M. Jardim, Asymptotic behaviour and the moduli spaceof doubly-periodic instantons, J. Eur. Math. Soc. (JEMS), 3 (2001), pp. 335–375.

[42] C. Birkenhake and H. Lange, Complex abelian varieties, Grundlehrender Mathematischen Wissenschaften, vol. 302, Springer-Verlag, Berlin, sec-ond ed., 2004.

[43] J. Bismut and D. Freed, The analysis of elliptic families. I. Metrics andconnections on determinant bundles, Commun. Math. Phys., 106 (1986),pp. 159–176.

[44] F. A. Bogomolov, Holomorphic tensors and vector bundles on projectivemanifolds, Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), pp. 1227–1287. En-glish transl. in Math. USSR-Izv. 13 (1979), no. 3, pp. 499–555.

[45] A. I. Bondal and M. M. Kapranov, Representable functors, Serre func-tors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat., 53 (1989),pp. 1183–1205. English transl. in Math. USSR-Izv., 35 (1990), pp. 519–541.

[46] , Enhanced triangulated categories, Mat. Sb., 181 (1990), pp. 669–683.English transl. in Math. USSR Sbornik, 70 (1991), pp. 93–107.

[47] A. I. Bondal, M. Larsen, and V. A. Lunts, Grothendieck ring of pre-triangulated categories, Int. Math. Res. Not., (2004), pp. 1461–1495.

[48] A. I. Bondal and D. O. Orlov, Semi orthogonal decomposition for alge-braic varieties. MPIM Preprint 95/15 (1995), also arXiv:math.AG/9506012.

[49] , Reconstruction of a variety from the derived category and groups ofautoequivalences, Compositio Math., 125 (2001), pp. 327–344.

[50] , Derived categories of coherent sheaves, in Proceedings of the Interna-tional Congress of Mathematicians (Beijing 2002). Vol. II, Higher Ed. Press,Beijing, 2002, pp. 47–56.

References 401

[51] A. I. Bondal and A. E. Polishchuk, Homological properties of asso-ciative algebras: the method of helices, Izv. Ross. Akad. Nauk Ser. Mat., 57(1993), pp. 3–50. English transl. in Russian Acad. Sci. Izv. Math., 42 (1994),pp. 219–260.

[52] A. I. Bondal and M. van den Bergh, Generators and representabilityof functors in commutative and noncommutative geometry, Mosc. Math. J.,3 (2003), pp. 1–36.

[53] J. Bonsdorff, A fourier transformation for Higgs bundles., J. reine angew.Math., 591 (2006), pp. 21–48.

[54] B. Booss and D. D. Bleecker, Topology and analysis. The Atiyah-Singerindex formula and gauge-theoretic physics, Universitext, Springer-Verlag,New York, 1985.

[55] C. Borcea, Diffeomorphisms of a K3 surface, Math. Ann., 275 (1986),pp. 1–4.

[56] R. Bott, On a theorem of Lefschetz, Michigan Math. J., 6 (1959), pp. 211–216.

[57] P. J. Braam and P. van Baal, Nahm’s transformation for instantons,Comm. Math. Phys., 122 (1989), pp. 267–280.

[58] T. Bridgeland, Spaces of stability conditions. Preprint arXiv:math/0611510.

[59] , Stability conditions and kleinian singularities. Preprint arXiv:math/0508257.

[60] , Fourier-Mukai transforms for elliptic surfaces, J. reine angew. Math.,498 (1998), pp. 115–133.

[61] , Equivalences of triangulated categories and Fourier-Mukai transforms,Bull. London Math. Soc., 31 (1999), pp. 25–34.

[62] , Flops and derived categories, Invent. Math., 147 (2002), pp. 613–632.

[63] , Derived categories of coherent sheaves, in International Congress ofMathematicians (Madrid 2006). Vol. II, Eur. Math. Soc., Zurich, 2006,pp. 563–582.

[64] , Stability conditions on a non-compact Calabi-Yau threefold, Comm.Math. Phys., 266 (2006), pp. 715–733.

[65] , Stability conditions on triangulated categories, Ann. of Math. (2), 166(2007), pp. 317–345.

402 References

[66] , Stability conditions on K3 surfaces, Duke Math. J., 141 (2008),pp. 241–291.

[67] T. Bridgeland and S. Iyengar, A criterion for regularity of local rings,C. R. Math. Acad. Sci. Paris, 342 (2006), pp. 723–726.

[68] T. Bridgeland, A. King, and M. Reid, The McKay correspondence asan equivalence of derived categories, J. Amer. Math. Soc., 14 (2001), pp. 535–554.

[69] T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for quotientvarieties. Preprint arXiv:math.AG/9811101.

[70] , Complex surfaces with equivalent derived categories, Math. Z., 236(2001), pp. 677–697.

[71] , Fourier-Mukai transforms for K3 and elliptic fibrations, J. AlgebraicGeom., 11 (2002), pp. 629–657.

[72] T. Bridgeland and V. Toledano Laredo, Stability conditions andStokes factors. Preprint arXiv:0801.3974.

[73] U. Bruzzo and A. Maciocia, Hilbert schemes of points on some K3 sur-faces and Gieseker stable bundles, Math. Proc. Cambridge Philos. Soc., 120(1996), pp. 255–261.

[74] U. Bruzzo, G. Marelli, and F. Pioli, A Fourier transform for sheaveson real tori. I. The equivalence Sky(T ) ' Loc(T ), J. Geom. Phys., 39 (2001),pp. 174–182.

[75] , A Fourier transform for sheaves on real tori. II. Relative theory, J.Geom. Phys., 41 (2002), pp. 312–329.

[76] U. Bruzzo and F. Pioli, Complex Lagrangian embeddings of moduli spacesof vector bundles, Differential Geom. Appl., 14 (2001), pp. 151–156. Erratum,ibid., 15 (2002), 201.

[77] N. Buchdahl, On compact Kahler surfaces, Ann. Inst. Fourier., 49 (1999),pp. 287–302.

[78] I. Burban and Y. Drozd, Coherent sheaves on rational curves with sim-ple double points and transversal intersections, Duke Math. J., 121 (2004),pp. 189–229.

[79] I. Burban and B. Kreußler, Derived categories of irreducible projectivecurves of arithmetic genus one, Compos. Math., 142 (2006), pp. 1231–1262.

References 403

[80] , On a relative Fourier-Mukai transform on genus one fibrations,Manuscripta Math., 120 (2006), pp. 283–306.

[81] D. Burns, Jr. and M. Rapoport, On the Torelli problem for kahlerianK-3 surfaces, Ann. Sci. Ecole Norm. Sup. (4), 8 (1975), pp. 235–273.

[82] A. Caldararu, Derived categories of twisted sheaves on elliptic threefolds,J. reine angew. Math., 544 (2002), pp. 161–179.

[83] , Nonfine moduli spaces of sheaves on K3 surfaces, Int. Math. Res.Not., (2002), pp. 1027–1056.

[84] J. W. S. Cassels, Rational quadratic forms, London Mathematical SocietyMonographs, vol. 13, Academic Press, London, 1978.

[85] B. Charbonneau, From spatially periodic instantons to singularmonopoles, Commun. Anal. Geom., 14 (2006), pp. 1–32.

[86] B. Charbonneau and J. Hurtubise, The Nahm transform for calorons,in Proceedings of the Geometry Conference in Honour of Nigel Hitchin, J.-P.Bourguignon, O. Garcıa-Prada, and S. Salamon, eds., Oxford, 2008, OxfordUniversity Press (in preparation).

[87] J.-C. Chen, Flops and equivalences of derived categories for threefolds withonly terminal Gorenstein singularities, J. Differential Geom., 61 (2002),pp. 227–261.

[88] S. Cherkis and A. Kapustin, Nahm transform for periodic monopoles andN = 2 super Yang-Mills theory, Comm. Math. Phys., 218 (2001), pp. 333–371.

[89] B. Conrad, Grothendieck duality and base change, Lecture Notes in Math-ematics, vol. 1750, Springer-Verlag, Berlin, 2000.

[90] E. Corrigan and P. Goddard, Construction of instanton and monopolesolutions and reciprocity, Ann. Physics, 154 (1984), pp. 253–279.

[91] M. Demazure, La place des surfaces K3 dans la classification des surfaces,in Geometrie des surfaces K3: modules et periodes (Palaiseau, 1981/1982),Asterisque, vol. 126, Societe Mathematique de France, Paris, 1985, pp. 29–38.

[92] D.-E. Diaconescu, D-branes, monopoles and Nahm equations, NuclearPhys. B, 503 (1997), pp. 220–238.

[93] I. V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J.Math. Sci., 81 (1996), pp. 2599–2630.

404 References

[94] R. Donagi, Spectral covers, in Current topics in complex algebraic geometry(Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., vol. 28, CambridgeUniv. Press, Cambridge, 1995, pp. 65–86.

[95] R. Donagi, A. Lukas, B. A. Ovrut, and D. Waldram, Holomor-phic vector bundles and non-perturbative vacua in M-theory, J. High EnergyPhys., 6 (1999), pp. 46, paper 34 (electronic).

[96] R. Donagi, B. A. Ovrut, T. Pantev, and D. Waldram, Standardmodels from heterotic M-theory, Adv. Theor. Math. Phys., 5 (2001), pp. 93–137.

[97] R. Donagi and T. Pantev, Torus fibrations, gerbes, and duality, Mem.Amer. Math. Soc., 193, No. 901 (2008).

[98] S. K. Donaldson, Nahm’s equations and the classification of monopoles,Comm. Math. Phys., 96 (1984), pp. 387–407.

[99] , Anti self-dual Yang-Mills connections over complex algebraic surfacesand stable vector bundles, Proc. London Math. Soc. (3), 50 (1985), pp. 1–26.

[100] , Infinite determinants, stable bundles and curvature, Duke Math. J.,54 (1987), pp. 231–247.

[101] , Polynomial invariants for smooth four-manifolds, Topology, 29 (1990),pp. 257–315.

[102] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, The Clarendon Press – Oxford University Press, Oxford, 1990.

[103] M. R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math.Phys., 42 (2001), pp. 2818–2843.

[104] , Dirichlet branes, homological mirror symmetry, and stability, in Pro-ceedings of the International Congress of Mathematicians (Beijing 2002).Vol. III, Higher Ed. Press, Beijing, 2002, pp. 395–408.

[105] V. Drinfeld, DG quotients of DG categories, J. Algebra, 272 (2004),pp. 643–691.

[106] W. G. Dwyer and J. Spalinski, Homotopy theories and model cate-gories, in Handbook of algebraic topology, North-Holland, Amsterdam, 1995,pp. 73–126.

[107] E. G. Evans and P. Griffith, Syzygies, London Mathematical SocietyLecture Note Series, vol. 106, Cambridge University Press, Cambridge, 1985.

References 405

[108] R. Fahlaoui and Y. Laszlo, Transformee de Fourier et stabilite sur lessurfaces abeliennes, Compositio Math., 79 (1991), pp. 271–278.

[109] D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds,Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, second ed., 1991.

[110] R. Friedman, Vector bundles and SO(3)-invariants for elliptic surfaces, J.Amer. Math. Soc., 8 (1995), pp. 29–139.

[111] , Algebraic surfaces and holomorphic vector bundles, Universitext,Springer-Verlag, New York, 1998.

[112] R. Friedman and J. W. Morgan, Smooth four-manifolds and complexsurfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 27,Springer-Verlag, Berlin, 1994.

[113] R. Friedman, J. W. Morgan, and E. Witten, Vector bundles and Ftheory, Comm. Math. Phys., 187 (1997), pp. 679–743.

[114] , Vector bundles over elliptic fibrations, J. Algebraic Geom., 8 (1999),pp. 279–401.

[115] K. Fukaya, Morse homotopy, A∞-category, and Floer homologies, in Pro-ceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993),Lecture Notes Ser., vol. 18, Seoul Nat. Univ., Seoul, 1993, pp. 1–102.

[116] , Floer homology and mirror symmetry. I, in Winter School on MirrorSymmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA,1999), AMS/IP Stud. Adv. Math., vol. 23, Amer. Math. Soc., Providence,RI, 2001, pp. 15–43.

[117] , Floer homology and mirror symmetry. II, in Minimal surfaces, geo-metric analysis and symplectic geometry (Baltimore, MD, 1999), Adv. Stud.Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 31–127.

[118] , Mirror symmetry of abelian varieties and multi-theta functions, J.Algebraic Geom., 11 (2002), pp. 393–512.

[119] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrerGrenzgebiete (3), vol. 2, Springer-Verlag, Berlin, second ed., 1998.

[120] P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 90 (1962),pp. 323–448.

406 References

[121] M. Garcıa Perez, A. Gonzalez-Arroyo, C. Pena, and P. van Baal,Nahm dualities on the torus — a synthesis, Nuclear Phys. B, 564 (2000),pp. 159–181.

[122] O. Garcıa-Prada, D. Hernandez Ruiperez, F. Pioli, and

C. Tejero Prieto, Fourier-Mukai and Nahm transforms for holomorphictriples on elliptic curves, J. Geom. Phys., 55 (2005), pp. 353–384.

[123] J. F. Glazebrook, M. Jardim, and F. W. Kamber, A Fourier-Mukaitransform for real torus bundles, J. Geom. Phys., 50 (2004), pp. 360–392.

[124] R. Godement, Topologie algebrique et theorie des faisceaux, Hermann,Paris, third ed., 1973.

[125] C. Gomez, R. Hernandez, and E. Lopez, K3 fibrations and softly brokenN = 4 supersymmetric gauge theories, Nuclear Phys. B, 501 (1997), pp. 109–133.

[126] T. L. Gomez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci., 111(2001), pp. 1–31.

[127] G. Gonzalez-Sprinberg and J.-L. Verdier, Construction geometriquede la correspondance de McKay, Ann. Sci. Ecole Norm. Sup. (4), 16 (1983),pp. 409–449.

[128] A. L. Gorodentsev, S. A. Kuleshov, and A. N. Rudakov, t-stabilitiesand t-structures on triangulated categories, Izv. Ross. Akad. Nauk Ser. Mat.,68 (2003), pp. 117–150. English transl. in Izv. Math. 68 (2004), No. 4, pp.749-781.

[129] L. Gottsche and D. Huybrechts, Hodge numbers of moduli spaces ofstable bundles on K3 surfaces, Internat. J. Math., 7 (1996), pp. 359–372.

[130] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Mod-ulraume komplexer Strukturen, Inst. Hautes Etudes Sci. Publ. Math., 5(1960), pp. 5–64. Berichtigung, ibid., 16 (1963), p. 35-36.

[131] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley& Sons Inc., New York, 1994.

[132] M. Gross, Special Lagrangian fibrations. I. Topology, and II. Geometry, Asurvey of techniques in the study of special Lagrangian fibrations, in WinterSchool on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds(Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., vol. 23, Amer. Math.Soc., Providence, RI, 2001, pp. 65–93 and 95–150.

References 407

[133] A. Grothendieck, Sur quelques points d’algebre homologique, TohokuMath. J., (1957), pp. 119–221.

[134] , Elements de geometrie algebrique. III. Etude cohomologique des fais-ceaux coherents. I, Inst. Hautes Etudes Sci. Publ. Math., (1961), p. 167.

[135] , Fondements de la geometrie algebrique. [Extraits du Seminaire Bour-baki, 1957–1962.], Secretariat mathematique, Paris, 1962.

[136] , Elements de geometrie algebrique. III. Etude cohomologique des fais-ceaux coherents. II, Inst. Hautes Etudes Sci. Publ. Math., (1963), p. 91.

[137] D. Happel, I. Reiten, and S. O. Smalø, Tilting in abelian categories andquasitilted algebras, Mem. Amer. Math. Soc., 120, No. 575 (1996), pp. viii+88.

[138] R. Hartshorne, Ample vector bundles, Inst. Hautes Etudes Sci. Publ.Math., (1966), pp. 63–94.

[139] , Residues and duality, With an appendix by P. Deligne. Lecture Notesin Mathematics, vol. 20, Springer-Verlag, Berlin, 1966.

[140] , Local cohomology, A seminar given by Grothendieck, Harvard Uni-versity, Fall 1961. Lecture Notes in Mathematics, vol. 41, Springer-Verlag,Berlin, 1967.

[141] , Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York, 1977.

[142] G. Hein and D. Ploog, Fourier-Mukai transforms and stable bundles onelliptic curves, Beitrage Algebra Geom., 46 (2005), pp. 423–434.

[143] D. Hernandez Ruiperez, A. C. Lopez Martın, and F. Sancho de

Salas, Relative integral functors for singular fibrations and singular part-ners, J. Eur. Math. Soc., to appear. Also arXiv:math/0610319.

[144] , Fourier-Mukai transform for Gorenstein schemes, Adv. Math., 211(2007), pp. 594–620.

[145] D. Hernandez Ruiperez and J. M. Munoz Porras, Stable sheaves onelliptic fibrations, J. Geom. Phys., 43 (2002), pp. 163–183.

[146] N. J. Hitchin, On the construction of monopoles, Comm. Math. Phys., 89(1983), pp. 145–190.

[147] N. J. Hitchin, A. Karlhede, U. Lindstrom, and M. Rocek, Hy-perkahler metrics and supersymmetry, Commun. Math. Phys., 108 (1987),pp. 535–589.

408 References

[148] R. P. Horja, Derived category automorphisms from mirror symmetry, DukeMath. J., 127 (2005), pp. 1–34.

[149] S. Hosono, B. H. Lian, K. Oguiso, and S.-T. Yau, Kummer structureson K3 surface: an old question of T. Shioda, Duke Math. J., 120 (2003),pp. 635–647.

[150] , Fourier-Mukai number of a K3 surface, in Algebraic structures andmoduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Prov-idence, RI, 2004, pp. 177–192.

[151] B. Hunt and R. Schimmrigk, Heterotic gauge structure of type II K3fibrations, Phys. Lett. B, 381 (1996), pp. 427–436.

[152] J. Hurtubise and M. K. Murray, On the construction of monopoles forthe classical groups, Comm. Math. Phys., 122 (1989), pp. 35–89.

[153] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, OxfordMathematical Monographs, The Clarendon Press – Oxford University Press,Oxford, 2006.

[154] , Derived and abelian equivalence of K3 surfaces, J. Algebraic Geom.,17 (2008), pp. 375–400.

[155] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves,Vieweg & Sohn, Braunschweig, 1997.

[156] D. Huybrechts, E. Macrı, and P. Stellari, Derived equivalences ofK3 surfaces and orientation. Preprint arXiv:0710.1645.

[157] , Stability conditions for generic K3 categories, Comp. Math., 144(2008), pp. 134–162.

[158] L. Illusie, Existence de resolutions globales, Expose II, in Theorie des in-tersections et theoreme de Riemann-Roch (SGA 6), Lecture Notes in Math-ematics, vol. 225, Springer-Verlag, Berlin, 1971, pp. 160–221.

[159] M. Inaba, Moduli of stable objects in a triangulated category. PreprintarXiv:math/0612078.

[160] , Toward a definition of moduli of complexes of coherent sheaves on aprojective scheme, J. Math. Kyoto Univ., 42 (2002), pp. 317–329.

[161] A. Ishii, K. Ueda, and H. Uehara, Stability conditions and An-singularities. Preprint arXiv:math/0609551.

[162] M. Itoh, Yang-Mills connections and the index bundles, Tsukuba J. Math.,13 (1989), pp. 423–441.

References 409

[163] M. Jardim, Construction of doubly-periodic instantons, Comm. Math.Phys., 216 (2001), pp. 1–15.

[164] , Classification and existence of doubly-periodic instantons, Q. J. Math.,53 (2002), pp. 431–442.

[165] , Nahm transform and spectral curves for doubly-periodic instantons,Comm. Math. Phys., 225 (2002), pp. 639–668.

[166] M. Jardim and A. Maciocia, A Fourier-Mukai approach to spectral datafor instantons, J. reine angew. Math., 563 (2003), pp. 221–235.

[167] D. Joyce, Holomorphic generating functions for invariants counting coher-ent sheaves on Calabi-Yau 3-folds, Geom. Topol., 11 (2007), pp. 667–725.

[168] , Motivic invariants of Artin stacks and ‘stack functions’, Q. J. Math.,58 (2007), pp. 345–392.

[169] , Configurations in abelian categories. IV. Invariants and changing sta-bility conditions, Adv. Math., 217 (2008), pp. 125–204.

[170] H. Kajiura, K. Saito, and A. Takahashi, Matrix factorization andrepresentations of quivers. II. Type ADE case, Adv. Math., 211 (2007),pp. 327–362.

[171] A. Kapustin, A. Kuznetsov, and D. Orlov, Noncommutative instan-tons and twistor transform, Comm. Math. Phys., 221 (2001), pp. 385–432.

[172] A. Kapustin and S. Sethi, The Higgs branch of impurity theories, Adv.Theor. Math. Phys., 2 (1998), pp. 571–591.

[173] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren derMathematischen Wissenschaften, vol. 292, Springer-Verlag, Berlin, 1990.

[174] P. Kaste, W. Lerche, C. A. Lutken, and J. Walcher, D-branes onK3-fibrations, Nuclear Phys. B, 582 (2000), pp. 203–215.

[175] Y. Kawamata, D-equivalence and K-equivalence, J. Differential Geom., 61(2002), pp. 147–171.

[176] , Equivalences of derived categories of sheaves on smooth stacks, Amer.J. Math., 126 (2004), pp. 1057–1083.

[177] B. Keller, Deriving DG categories, Ann. Sci. Ecole Norm. Sup. (4), 27(1994), pp. 63–102.

[178] , On differential graded categories, in International Congress of Mathe-maticians (Madrid 2006). Vol. II, Eur. Math. Soc., Zurich, 2006, pp. 151–190.

410 References

[179] G. M. Kelly, Chain maps inducing zero homology maps, Proc. CambridgePhilos. Soc., 61 (1965), pp. 847–854.

[180] G. Kempf, Toward the inversion of abelian integrals. I, Ann. of Math. (2),110 (1979), pp. 243–273.

[181] , Toward the inversion of abelian integrals. II, Amer. J. Math., 101(1979), pp. 184–202.

[182] A. King, Moduli of representations of finite-dimensional algebras, Quart. J.Math. Oxford Ser. (2), 45 (1994), pp. 515–530.

[183] A. Klemm, W. Lerche, and P. Mayr, K3-fibrations and heterotic–typeII string duality, Phys. Lett. B, 357 (1995), pp. 313–322.

[184] S. Kobayashi, Differential geometry of complex vector bundles, Publicationsof the Mathematical Society of Japan, vol. 15, Princeton University Press,Princeton, NJ, 1987.

[185] S. Kobayashi and K. Nomizu, Foundations of differential geometry. VolI, Interscience Publishers, 1963.

[186] K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2), 77(1963), 563–626; ibid., 78 (1963), pp. 1–40.

[187] J. Kollar and S. Mori, Birational geometry of algebraic varieties, Cam-bridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cam-bridge, 1998.

[188] M. Kontsevich, Homological algebra of mirror symmetry, in Proceedingsof the International Congress of Mathematicians (Zurich, 1994). Vol. 1,Birkhauser, Basel, 1995, pp. 120–139.

[189] P. B. Kronheimer, The construction of ALE spaces as hyper-Kahler quo-tients, J. Differential Geom., 29 (1989), pp. 665–683.

[190] P. B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALEgravitational instantons, Math. Ann., 288 (1990), pp. 263–307.

[191] S. A. Kuleshov, A theorem on the existence of exceptional bundles onsurfaces of type K3, Math. USSR-Izv., 34 (1990), pp. 373–388.

[192] A. Lamari, Courants kahleriens et surfaces compactes, Ann. Inst. Fourier(Grenoble), 49 (1999), pp. vii, x, 263–285.

[193] G. Laumon and L. Moret-Bailly, Champs algebriques, Ergebnisse derMathematik und ihrer Grenzgebiete (3), vol. 39, Springer-Verlag, Berlin,2000.

References 411

[194] H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, PrincetonMathematical Series, vo. 38, Princeton University Press, Princeton, NJ,1989.

[195] N. C. Leung, S.-T. Yau, and E. Zaslow, From special Lagrangianto Hermitian-Yang-Mills via Fourier-Mukai transform, Adv. Theor. Math.Phys., 4 (2000), pp. 1319–1341.

[196] Y. Li, Spectral curves, theta divisors and Picard bundles, Internat. J. Math.,2 (1991), pp. 525–550.

[197] M. Lieblich, Moduli of complexes on a proper morphism, J. AlgebraicGeom., 15 (2006), pp. 175–206.

[198] A. C. Lopez Martın, Simpson Jacobians of reducible curves, J. reineangew. Math., 582 (2005), pp. 1–39.

[199] , Relative Jacobians of elliptic fibrations with reducible fibers, J. Geom.Phys., 56 (2006), pp. 375–385.

[200] M. Lubke and A. Teleman, The Kobayashi-Hitchin correspondence,World Scientific, Singapore, 1995.

[201] S. Mac Lane, Categories for the working mathematician, Graduate Textsin Mathematics, vol. 5, Springer-Verlag, New York, second ed., 1998.

[202] A. Maciocia, Generalized Fourier-Mukai transforms, J. reine angew.Math., 480 (1996), pp. 197–211.

[203] , Gieseker stability and the Fourier-Mukai transform for abelian sur-faces, Quart. J. Math. Oxford Ser. (2), 47 (1996), pp. 87–100.

[204] E. Macrı, Some examples of moduli spaces of stability conditions on derivedcategories. Preprint arXiv:math.AG/0411613.

[205] , Stability conditions on curves, Math. Res. Lett., 14 (2007), pp. 657–672.

[206] E. Macrı, S. Mehrotra, and P. Stellari, Inducing stability conditions.Preprint arXiv:0705.3752.

[207] B. Malgrange, Ideals of differentiable functions, Tata Institute of Funda-mental Research Studies in Mathematics, vol. 3, Oxford University Press,Bombay, 1967.

[208] M. Mamone Capria and S. M. Salamon, Yang-Mills fields on quater-nionic spaces, Nonlinearity, 1 (1988), pp. 517–530.

412 References

[209] Y. I. Manin, Correspondences, motifs and monoidal transformations, Mat.Sb. (N.S.), 77 (119) (1968), pp. 475–507. English transl. in Math. USSR-Sb.6 (1968), pp. 439–470.

[210] E. Markman, Brill-Noether duality for moduli spaces of sheaves on K3surfaces, J. Algebraic Geom., 10 (2001), pp. 623–694.

[211] M. Maruyama, Moduli of stable sheaves. I, J. Math. Kyoto Univ., 17 (1977),pp. 91–126.

[212] , Moduli of stable sheaves. II, J. Math. Kyoto Univ., 18 (1978), pp. 557–614.

[213] K. Matsuki and R. Wentworth, Mumford–Thaddeus principle on themoduli space of vector bundles on an algebraic surface, Internat. J. Math., 8(1997), pp. 97–148.

[214] H. Matsumura, Commutative algebra, Mathematics Lecture Note Series,vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., sec-ond ed., 1980.

[215] , Commutative ring theory, Cambridge Studies in Advanced Mathe-matics, vol. 8, Cambridge University Press, Cambridge, second ed., 1989.

[216] S. Meinhardt, Stability conditions on generic tori. Preprint arXiv:0708.3053.

[217] S. Meinhardt and H. Partsch, Quotient categories, stability conditions,and birational geometry. Preprint arXiv:0805.0492.

[218] A. Mellit and S. Okada, Joyce invariants for K3 surfaces and mocktheta functions. Preprint arXiv:0802.3136.

[219] R. Miranda, Smooth models for elliptic threefolds, in The birational geom-etry of degenerations (Cambridge, Mass., 1981), Progress in Mathematics,vol. 29, Birkhauser, Boston, MA, 1983, pp. 85–133.

[220] , The basic theory of elliptic surfaces, ETS Editrice, Pisa, 1989.

[221] S. Mori, Flip theorem and the existence of minimal models for 3-folds, J.Amer. Math. Soc., 1 (1988), pp. 117–253.

[222] D. R. Morrison, The geometry underlying mirror symmetry, in New trendsin algebraic geometry (Warwick, 1996), London Math. Soc. Lecture NoteSer., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 283–310.

[223] D. R. Morrison and C. Vafa, Compactifications of F -theory on Calabi-Yau threefolds. II, Nuclear Phys. B, 476 (1996), pp. 437–469.

References 413

[224] S. Mukai, Duality between D(X) and D(X) with its application to Picardsheaves, Nagoya Math. J., 81 (1981), pp. 153–175.

[225] , Symplectic structure of the moduli space of sheaves on an abelian orK3 surface, Invent. Math., 77 (1984), pp. 101–116.

[226] , Fourier functor and its application to the moduli of bundles on anabelian variety, in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math.,vol. 10, North-Holland, Amsterdam, 1987, pp. 515–550.

[227] , On the moduli space of bundles on K3 surfaces. I, in Vector bundles onalgebraic varieties (Bombay, 1984), Tata Institute of Fundamental ResearchStudies in Mathematics, vol. 11, Oxford University Press, Bombay, 1987,pp. 341–413.

[228] , Duality of polarized K3 surfaces, in New trends in algebraic geometry(Warwick, 1996), London Math. Soc. Lecture Note Ser., vol. 264, CambridgeUniv. Press, Cambridge, 1999, pp. 311–326.

[229] D. Mumford, Abelian varieties, Tata Institute of Fundamental ResearchStudies in Mathematics, vol. 5, Oxford University Press, Bombay, 1970.

[230] W. Nahm, Self-dual monopoles and calorons, in Group theoretical methodsin physics (Trieste, 1983), Lecture Notes in Phys., vol. 201, Springer-Verlag,Berlin, 1984, pp. 189–200.

[231] H. Nakajima, Monopoles and Nahm’s equations, in Einstein metrics andYang-Mills connections (Sanda, 1990), Lecture Notes in Pure and Appl.Math., vol. 145, Dekker, New York, 1993, pp. 193–211.

[232] I. Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom.,10 (2001), pp. 757–779.

[233] J. C. Naranjo, Fourier transform and Prym varieties, J. reine angew.Math., 560 (2003), pp. 221–230.

[234] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniquesand Brown representability, J. Amer. Math. Soc., 9 (1996), pp. 205–236.

[235] V. V. Nikulin, Integer symmetric bilinear forms and some of their geomet-ric applications, Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), pp. 111–177,238. English transl. in Math. USSR-Izv., 14 (1979), no. 1, pp. 103-167.

[236] T. Nitta, Vector bundles over quaternionic Kahler manifolds, TohokuMath. J. (2), 40 (1988), pp. 425–440.

414 References

[237] T. M. W. Nye, The geometry of calorons, PhD thesis, University of Edin-burgh, 2001. Also arXiv:hep-th/0311215.

[238] K. G. O’Grady, The weight-two Hodge structure of moduli spaces ofsheaves on a K3 surface, J. Algebraic Geom., 6 (1997), pp. 599–644.

[239] K. Oguiso, K3 surfaces via almost-primes, Math. Res. Lett., 9 (2002),pp. 47–63.

[240] S. Okada, On stability manifolds of Calabi-Yau surfaces, Int. Math. Res.Not., 2006 (2006), pp. 16, art. ID 58743.

[241] , Stability manifold of P1, J. Algebraic Geom., 15 (2006), pp. 487–505.

[242] D. O. Orlov, Equivalences of derived categories and K3 surfaces, J. Math.Sci. (New York), 84 (1997), pp. 1361–1381.

[243] , Derived categories of coherent sheaves on abelian varieties and equiv-alences between them, Izv. Ross. Akad. Nauk Ser. Mat., 66 (2002), pp. 131–158. English transl. Izv. Math. 66 (2002), no. 3, pp. 569–594.

[244] R. Pandharipande and R. Thomas, Curve counting via stable pairs inthe derived category. Preprint arXiv:0707.2348.

[245] , Stable pairs and BPS invariants. Preprint arXiv:0711.3899.

[246] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie lo-cale. Applications a la demonstration de conjectures de M. Auslander, H.Bass et A. Grothendieck, Inst. Hautes Etudes Sci. Publ. Math., 42 (1973),pp. 47–119.

[247] F. Pioli, Funtori integrali e fasci di Picard su varieta jacobiane e di Prym,PhD thesis, Universita di Genova, 1999.

[248] I. I. Pjateckiı-Sapiro and I. R. Safarevic, Torelli’s theorem for al-gebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971),pp. 530–572.

[249] D. Ploog, Groups of autoequivalences of derived categories of smooth pro-jective varieties, PhD thesis, Freie Universitat Berlin, 2005.

[250] , Equivariant autoequivalences for finite group actions, Adv. Math., 216(2007), pp. 62–74.

[251] A. Polishchuk, Abelian varieties, theta functions and the Fourier trans-form, Cambridge Tracts in Mathematics, vol. 153, Cambridge UniversityPress, Cambridge, 2003.

References 415

[252] , Classification of holomorphic vector bundles on noncommutative two-tori, Doc. Math., 9 (2004), pp. 163–181 (electronic).

[253] , Constant families of t-structures on derived categories of coherentsheaves, Mosc. Math. J., 7 (2007), pp. 109–134.

[254] A. Polishchuk and A. Schwarz, Categories of holomorphic vector bun-dles on noncommutative two-tori, Comm. Math. Phys., 236 (2003), pp. 135–159.

[255] Z. Qin, Moduli of simple rank-2 sheaves on K3-surfaces, Manuscripta Math.,79 (1993), pp. 253–265.

[256] D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol.43, Springer-Verlag, Berlin, 1967.

[257] M. Reid, Young person’s guide to canonical singularities, in Algebraic geom-etry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math.,vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414.

[258] D. S. Rim, Formal deformation theory, in Groupes de monodromie engeometrie algebrique. I, Seminaire de Geometrie Algebrique du Bois-Marie1967–1969 (SGA 7 I), Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972, pp. 32–132.

[259] P. Roberts, Intersection theorems, in Commutative algebra (Berkeley, CA,1987), Math. Sci. Res. Inst. Publ., vol. 15, Springer-Verlag, New York, 1989,pp. 417–436.

[260] R. Rouquier, Categories derivees et geometrie birationnelle (d’apres Bon-dal, Orlov, Bridgeland, Kawamata et al.), in Seminaire Bourbaki. Volume2004/2005. Exposes 938–951, Asterisque, vol. 307, Societe Mathematique deFrance, Paris, 2006, pp. 283–307.

[261] A. Rudakov, Stability for an abelian category, J. Algebra, 197 (1997),pp. 231–245.

[262] S. Salamon, Quaternionic Kahler manifolds, Invent. Math., 67 (1982),pp. 143–171.

[263] H. Schenk, On a generalised Fourier transform of instantons over flat tori,Comm. Math. Phys., 116 (1988), pp. 177–183.

[264] R. L. E. Schwarzenberger, Jacobians and symmetric products, IllinoisJ. Math., 7 (1963), pp. 257–268.

416 References

[265] P. Seidel and R. Thomas, Braid group actions on derived categories ofcoherent sheaves, Duke Math. J., 108 (2001), pp. 37–108.

[266] J.-P. Serre, Algebre locale. Multiplicites, Cours au College de France, 1957–1958, redige par Pierre Gabriel, Lecture Notes in Mathematics, vol. 11,Springer-Verlag, Berlin, 1965.

[267] , A course in arithmetic, Graduate Texts in Mathematics, vol. 7,Springer-Verlag, New York, 1973.

[268] T. Shioda, The period map of Abelian surfaces, J. Fac. Sci. Univ. TokyoSect. IA Math., 25 (1978), pp. 47–59.

[269] C. T. Simpson, Moduli of representations of the fundamental group of asmooth projective variety. I, Inst. Hautes Etudes Sci. Publ. Math., 79 (1994),pp. 47–129.

[270] Y. T. Siu, Every K3 surface is Kahler, Invent. Math., 73 (1983), pp. 139–150.

[271] P. Stellari, Some remarks about the FM-partners of K3 surfaces withPicard numbers 1 and 2, Geom. Dedicata, 108 (2004), pp. 1–13.

[272] S. Szabo, Nahm transform for integrable connections on the Riemannsphere, in Memoires de la SMF, Soc. Math. France, Paris, 2007.

[273] B. Szendroi, Diffeomorphisms and families of Fourier-Mukai transformsin mirror symmetry, in Applications of algebraic geometry to coding the-ory, physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys.Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 317–337.

[274] G. Tabuada, Une structure de categorie de modeles de Quillen sur lacategorie des dg-categories, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 15–19.

[275] A. Takahashi, Matrix factorizations and representations of quivers. I.Preprint arXiv:math/0506347.

[276] C. Tejero Prieto, Fourier-Mukai transform and adiabatic curvature ofspectral bundles for Landau Hamiltonians on Riemann surfaces, Comm.Math. Phys., 265 (2006), pp. 373–396.

[277] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds,and bundles on K3 fibrations, J. Differential Geom., 54 (2000), pp. 367–438.

[278] , Stability conditions and the braid group, Comm. Anal. Geom., 14(2006), pp. 135–161.

References 417

[279] Y. Toda, Limit stable objects on Calabi-Yau 3-folds. Preprint arXiv:0803.2356.

[280] , Birational Calabi-Yau threefolds and BPS state counting, Commun.Number Theory Phys., 2 (2008), pp. 63–112.

[281] , Moduli stacks and invariants of semistable objects on K3 surfaces,Adv. Math., 217 (2008), pp. 2736–2781.

[282] Y. Toda, Stability conditions and crepant small resolutions, Trans. Amer.Math. Soc., 360 (2008), pp. 6149–6178.

[283] Y. Toda, Stability conditions and Calabi-Yau fibrations, J. AlgebraicGeom., 18 (2009), pp. 101–133.

[284] B. Toen, The homotopy theory of dg-categories and derived Morita theory,Invent. Math., 167 (2007), pp. 615–667.

[285] L. W. Tu, Semistable bundles over an elliptic curve, Adv. Math., 98 (1993),pp. 1–26.

[286] K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Millsconnections in stable vector bundles, Comm. Pure Appl. Math., 39 (1986),pp. S257–S293.

[287] P. van Baal, Instanton moduli for T 3 ×R, Nuclear Phys. B Proc. Suppl.,49 (1996), pp. 238–249.

[288] , Nahm gauge fields for the torus, Phys. Lett. B, 448 (1999), pp. 26–32.

[289] M. Van den Bergh, Three-dimensional flops and noncommutative rings,Duke Math. J., 122 (2004), pp. 423–455.

[290] M. Verbitsky, Hyperholomorphic bundles over a hyper-Kahler manifold,J. Algebraic Geom., 5 (1996), pp. 633–669.

[291] J.-L. Verdier, Dualite dans la cohomologie des espaces localement com-pacts, in Seminaire Bourbaki, vol. 9 (annees 1964/65-1965/66, exposes 277–312), Societe Mathematique de France, Paris, 1995, pp. 337–349.

[292] E. Witten, Small instantons in string theory, Nuclear Phys. B, 460 (1996),pp. 541–559.

[293] K. Yoshioka, Some notes on the moduli of stable sheaves on elliptic sur-faces, Nagoya Math. J., 154 (1999), pp. 73–102.

[294] , Moduli spaces of stable sheaves on abelian surfaces, Math. Ann., 321(2001), pp. 817–884.

418 References

[295] , Twisted stability and Fourier-Mukai transform. I, Compositio Math.,138 (2003), pp. 261–288.

[296] , Twisted stability and Fourier-Mukai transform. II, ManuscriptaMath., 110 (2003), pp. 433–465.

[297] K. Zuo, The moduli spaces of some rank-2 stable vector bundles over alge-braic K3-surfaces, Duke Math. J., 64 (1991), pp. 403–408.

Subject Index

1-irreducible, 159

Abelian

Fourier-Mukai transform, 85

schemes, 188

variety, 82

Adjoint

bundle, 149

functors, 12, 282

ALE spaces, 182

Algebraic group, 82

Ample sequences, 35

Associated bundle, 149

Atiyah-Ward correspondence, 176

Automorphy factor, 162

Base change

for integral functors, 8, 186

in derived category, 331

Beılinson resolution of the diagonal, 45

Bogomolov number, 223

Category, 281

k-bilinear, 284

Abelian, 285

additive, 283

derived, 297

of a dg-catgeory, 310

dg-derived of a dg-category, 311

differential graded, 307

essentially small, 363

homotopy, 289

large, 281

locally small, 281

numerically finite, 367

of complexes, 287

of dg-categories, 308

of finite length, 364

of finite type, 366

quasi-Abelian, 286

saturated, 15

small, 281, 363

triangulated, 306

Chern

character


relative, 190

connection, 153

Clifford bundle, 155

Cohomology

long exact sequence, 290

of a complex, 288

Complex, 287

double, 292

of finite homological dimension, 303

of finite Tor-dimension, 6, 326

of homomorphisms, 288

perfect, 3, 303

simple (associated with a double com-

plex), 292

tensor product, 293

Cone of a morphism of complexes, 289

Connection

on a principal bundle, 149

flat, 148

irreducible, 149

on a vector bundle, 148

reducible, 149

Convolution

of complexes, 40

of kernels, 5

420 Subject Index

Crepant

birational map, 235

morphism, 235

Curvature, 148

D-equivalence implies K-equivalence, 69

Decomposable triangulated category, 32

Derived

direct image, 317

homomorphism functor, 318

inverse image, 327

tensor product, 324

dg-category, 307

dg-derived, 311

dg-functor, 307

dg-module, 309

cofibrant, 310

fibrant, 310

Dirac

Laplacian, 159

operator, 155, 159

Discriminant group of a lattice, 252, 341

Dual Abelian variety, 83

Duality in derived categories, 63, 347

Elliptic

fibration, 190

surface, relatively minimal, 190

Equivalence of categories, 14, 282

Essential image of a functor, 300

Essentially small category, 363

Euler characteristic of two complexes, 4

Exact triangle in derived category, 303

Exponent, of an isogeny, 82

Finite

homological dimension of a complex,

303

Tor-dimension

of a complex, 6, 326

of a morphism, 328

Finiteness, of the number of Fourier-Mukai

partners of an algebraic surface, 242

Fitting ideal, 217, 355

Flat base change in derived category,

331

Fourier-Mukai

functor, 60

partners, 61

of a curve, 242

of a K3 surface, 249

of a Kummer surface, 250

of a nonminimal projective surface,

256

of a surface of Kodaira dimension

−∞ and not elliptic, 246


2, 245


1, 248

of an elliptic surface, 248

transform, 60

on Abelian schemes, 188

on Abelian varieties, 85

on K3 surfaces, 120

Fully faithful integral functors, 15

Functor, 282

k-bilinear, 285

additive, 284

cohomological, 290, 306

of finite type, 15

derived, 312

exact

full faithfulness of, 33

of Abelian categories, 286

of triangulated categories, 306

left derived, 313

of dg-categories, 307

representable, 282

right derived, 313, 314

Gauge group, 150

Genus, of a lattice, 341

Green function, 153

Harder-Narasimhan filtration, 363

Heart

of a stability condition, 363

of a t-structure, 264

Hilbert scheme of points, 142

Hitchin-Kobayashi correspondence, 153

Hodge

Subject Index 421

duality, 150

isometry, 115

Homogeneous

bundles on Abelian varieties, 90

sheaf (on a K3 surface), 132

Homological dimension of a complex, 336

finite, 303

Homomorphisms of Abelian varieties, 82

Homotopy category of a dg-category, 308

Hyperkahler manifold, 173

Indecomposable triangulated category, 32

Index of a singular variety, 235

Index theorem

(Atiyah-Singer), 156

for families, 157

Instanton, 150

Integral functor, 5

full faithfulness of, 20

relative, 8, 184

relative, for Weierstraß fibrations, 193

Isogeny, 82

Isometric isomorphism (of Abelian vari-

eties), 254

Isotropic embeddings of moduli spaces,

105

IT condition, 7, 168, 179

Jacobian of a Weierstraß fibration, 197

Jordan-Holder filtration, 365

K-equivalence, 68

K3 surface, 112

branched over a sextic, 113

Kahlerian, 114

Kummer, 113

quartic, 113

reflexive, 124

strongly reflexive, 125

Kunneth formula in derived category, 334

Kernel of an integral functor, 5

convolution of, 5

relative, 64

strongly simple, 19

uniqueness of, 51

Kodaira

conjecture, 114

dimension, 65

numerical dimension, 66

Kodaira-Spencer map, 17

for families of complexes, 259

Koszul

algebra, 46

line bundle, 47

Kummer surface, see K3 surface, Kum-

mer

Fourier-Mukai partners of a, 250

Lagrangian embeddings of moduli spaces,

105

Lattice, 339

E8, 340

genus of, 341

unimodular, 340

Length of a bounded complex, 336

Marking of a K3 surface, 115

Minimal surface, 244

Modified support, 214, 356

Moduli space

of instantons, 151

of K3 surfaces, 116

of semistable pure sheaves, 353

Morphism

of finite

Tor-dimension, 328

of functors, 282

Motivic invariant, 391

Mukai

pairing, 4

vector, 3

Neron-Severi group, 114, 162

Nahm transform, 158

Nodal curves, 114

Nondegenerate line bundles (on an Abelian

variety), 92

Normalization, of the Poincare bundle,

84

Numerical Kodaira dimension, 66

Numerically

effective line bundle, 65

422 Subject Index

equivalent, 267

Orbit space, 151

Orlov’s representability theorem, 44

Overlattice, 342

Parseval formula (preservation of the Ext

groups), 23

Period

domain, for K3 surfaces, 115

map, for K3 surfaces, 115

Perverse

point sheaf, 268

ideal sheaf, 267

sheaf, 265

structure sheaf, 267

Picard

functor, 83

lattice, 114

number, 114

sheaves, 95

Poincare

bundle, 83, 163

relative sheaf, 193

sheaf, 95

Polarization

of the dual Abelian variety, 94

on Abelian varieties, 94

principal, 94

Polarized rank, 352

Polystable sheaf, 154

Pontrjagin product, 89

Preservation

of the Ext groups, 23

Preservation of stability

absolute, for elliptic Calabi-Yau three-

folds, 228

absolute, for elliptic surfaces, 221

for Abelian surfaces, 102, 108

for elliptic curves, 98

for K3 surfaces, 139, 145, 380

for Weierstraß curves, 210

relative, for elliptic fibrations, 213

Primitive

embedding, 342

vector, 342

Principal bundle, 149

Projection formula in derived category,

330

Pure sheaves, 351

Quasi-Abelian category, 286

Quasi-universal sheaf, 118

Quaternionic

instanton, 175

Kahler manifold, 173

Reconstruction theorem, 70

Reduction of the structure group, 149

Reflexive K3 surfaces, 124

Relative

connection, 167

differential operator, 165

Dirac operators, 170

Dolbeault complex, 166

dualizing complex, 347

Fourier-Mukai functors, 64

Fourier-Mukai transforms, 64

on elliptic fibrations, 197

kernel, 64

Todd class of an elliptic fibration, 191

Relatively

semi-stable sheaf, 190

torsion-free sheaf, 190

Resolution of the diagonal

Beılinson’s, 45

Kawamata’s, 46

Resolution property, of an algebraic va-

riety, 321

S-equivalence, 353

Semi-orthogonal decomposition, 263

Semicharacter, 162

Semistable

bundles on elliptic curves, 97

object in a category, 364

Serre functor, 13

Shift functor, 288, 306

Signature of the intersection form, 114

Simpson stability, 351

Singularity

canonical, 235

Subject Index 423

terminal, 235

Skyscraper sheaves, 5

Small category, 363

Spanning class, 32

Special

sheaf, 72

complex, 72

Lagrangian submanifolds, 108

Spectral cover, 218

Spherical object, 78

Spin structure, 155

Stability condition, 363

algebraic, 388

good, 382

locally finite, 365

numerical, 367

Strict short exact sequence, 286

Strongly reflexive K3 surfaces, 125

Strongly simple

kernel, 19

sheaf, 22

Subcategory

right admissible, 263

right orthogonal, 263

thick (of an Abelian category), 300

thin, 371

triangulated, 306

Support of a complex, 335

Symplectic

form (on the moduli space of stable

sheaves), 104, 142

morphisms of moduli spaces, 105, 142

t-structure, 264

Theta divisor, 94

Tor-dimension, finite

of a complex, 6, 326

of a morphism, 328

Torelli theorem

for K3 surfaces, 115

weak, for K3 surfaces, 115

Transcendental lattice, 114

Triangulated

category, 306

subcategory, 306

Truncation


of a complex, 291

of a double complex, 293

Twist functor, 78

Twisted Gieseker stability, 108, 145

Twistor space, 174

Unipotent bundles, 86

Uniqueness of the kernel, 51

Universal

bundle on the instanton moduli space,

152

connection on the instanton moduli

space, 152

family, relative, 258

line bundle, 84

perverse point sheaf, 270

sheaf, 17, 75, 118, 119, 353

Walls of a space of stability conditions,

382

Weak Torelli theorem, for K3 surfaces,

115

Weierstraß fibration, 190

Weil-Petersson metric, 151

Weitzenbock formula, 159

WIT condition, 7, 168

Without flat factors, 164

Yoneda product, 322

Yoneda’s

formula, 320

lemma, 283

fourier-mukai and nahm transforms in geometry and mathematical physics

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