and geometry of · contents parti introductorylectures k3andenriquessurfaces 3 shigeyuki kondo 1...

Radu Laza • Matthias Schiitt • Noriko Yui

Editors

Arithmetic and Geometry of

K3 Surfaces and Calabi-Yau

Threefolds

The Fields Institute for Research

)S in the Mathematical Sciences ^ Springei

Contents

Part I Introductory Lectures

K3 and Enriques Surfaces 3

Shigeyuki Kondo

1 Introduction 3

2 Lattices 4

2.1 Definition 4

2.2 Examples 5

2.3 Unimodular Lattices 5

2.4 Proposition 5

2.5 Proposition 5

2.6 Discriminant Quadratic Form 6

2.7 Overlattices 6

2.8 Proposition 7

2.9 Examples 7

2.10 Primitive Embeddings 8

2.11 Proposition 8

2.12 Corollary 8

2.13 Example 9

2.14 Corollary 9

2.15 Corollary 9

2.16 Theorem ([30]) 10

3 Periods of K3 and Enriques Surfaces 10

3.1 Periods of K3 Surfaces 10

3.2 Periods of Enriques Surfaces 11

3.3 Remark 12

3.4 Remark 13

3.5 Example 14

4 Automorphisms 15

4.1 Torelli Type Theorem and the Group of Automorphismsfor an Algebraic K?> Surface 15

XV

XVI Contents

4.2 Theorem ([35]) 16

4.3 Theorem ([35]) 16

4.4 Corollary 17

4.5 The Leech Lattice and the Group of Automorphismsof a Generic Jacobian Kummer Surface 17

4.6 Theorem ([10], Chap. 27) 18

4.7 Proposition ([4]) 18

4.8 Finite Groups of Automorphisms of K3 Surfaces 19



4.11 Theorem ([28]) 20

4.12 Automorphisms of Enriques Surfaces 21

4.13 Theorem ([2, 31], Theorem 10.1.2) 21

5 Borcherds Products 22

5.1 Theorem ([7]) 23

5.2 Example ([5]) 23

5.3 Example ([6, 7]) 24

5.4 Example ([15, 25]) 25

References 27

Transcendental Methods in the Study of Algebraic Cycles with a Special

Emphasis on Calabi-Yau Varieties 29

James D. Lewis

1 Introduction 29

2 Notation 31

3 Some Hodge Theory 31

3.4 Formalism of Mixed Hodge Structures 33

4 Algebraic Cycles 36

4.6 Generalized Cycles 38

5 A Short Detour via Milnor A"-Theory 40

5.1 The Gersten-Milnor Complex 41

6 Hypercohomology 42

7 Delignc Cohomology 43

7.6 Alternate Take on Deligne Cohomology 46

7.8 Dcligne-Beilinson Cohomology 47

8 Examples of H'7~"'(X, K.*'x) and Corresponding Regulators 51

8.1 Case m = 0 and CY Threefolds 51

8.5 Deligne Cohomology and Normal Functions 55

8.9 Case m = 1 and K3 Surfaces 56

8.18 Torsion Indecomposables 59

8.21 Case m = 2 and Elliptic Curves 60

8.22 Constructing K2(X) Classes on Elliptic Curves X 61

References 68

Contents xvii

Two Lectures on the Arithmetic of K3 Surfaces 71

Matthias Schiitt

1 Introduction 71

2 Motivation: Rational Points on Algebraic Curves 72

3 K3 Surfaces and Rational Points 73

4 Elliptic K3 Surfaces 74

5 Picard Number One 76

6 Computation of Picard Numbers 77

7 K3 Surfaces of Picard Number One 79

7.1 van Luijk's Approach 80

7.2 Kloosterman's Improvement 80

7.3 Elsenhans-Jahnel's Work 80

7.4 Outlook 81

7.5 Feasibility 81

8 Hasse Principle for K3 Surfaces 82

9 Rational Curves on K3 Surfaces 83

10 Isogeny Notion for K3 Surfaces 83

11 Singular K3 Surfaces 84

11.1 Torelli Theorem for Singular K3 Surfaces 85

11.2 Surjectivity of the Period Map 86

11.3 Singular Abelian Surfaces 86

12 Shioda-Inose Structures 87

13 Mordell-Weil Ranks of Elliptic K3 Surfaces 89

14 Fields of Definition of Singular K3 Surfaces 91

14.1 Mordell-Weil Ranks Over Q 93

15 Modularity of Singular K3 Surfaces 93

References 96

Modularity of Calabi-Yau Varieties:

2011 and Beyond 101

Noriko Yui

1 Introduction 102

1.1 Brief History Since 2003 102

1.2 Plan of Lectures 102

1.3 Disclaimer 103

1.4 Calabi-Yau Varieties: Definition 103

2 The Modularity of Galois Representations of Calabi-Yau

Varieties (or Motives) Over Q 105

3 Results on Modularity of Galois Representations 110

3.1 Two-Dimensional Galois Representations Arisingfrom Calabi-Yau Varieties Over Q 110

3.2 Modularity of Higher Dimensional Galois

Representations Arising from K3 Surfaces Over Q 112

Will Contents

3.3 The Modularity of Higher Dimensional Galois

Representations Arising from Calabi-Yau Threefolds

OverQ 117

4 The Modularity of Mirror Maps of Calabi-Yau Varieties,

and Mirror Moonshine 126

5 The Modularity of Generating Functions of CountingSome Quantities on Calabi-Yau Varieties 129

6 Future Prospects 130

6.1 The Potential Modularity 130

6.2 The Modularity of Moduli of Families

of Calabi-Yau Varieties 131

6.3 Congruences, Formal Groups 131

6.4 The Griffiths Intermediate Jacobians

of Calabi-Yau Threefolds 131

6.5 Geometric Realization Problem

(the Converse Problem) 132

6.6 Modular Forms and Gromov-Witten Invariants 133

6.7 Automorphic Black Hole Entropy 133

6.8 /VKj-Moonshine 133

References 137

Part II Research Articles: Arithmetic and Geometry of K3, Enriquesand Other Surfaces

Explicit Algebraic Coverings of a Pointed Torus 143

Ane S.I. Anema and Jaap Top1 Introduction 143

2 The Coverings 144

3 The Proofs 145

3.1 2-Torsion 145

3.2 3-Torsion 146

3.3 f-Torsion with f > 5 147

4 Intermediate Coverings 150

4.1 All .v-Coordinates 150

4.2 One Point 151

4.3 One v-Coordinatc 151

References 152

Elliptic Fibrations on the Modular Surface Associated to T|(8) 153

M.J. Benin and O. Lecacheux

1 Introduction 153

2 Definitions 155

3 Discriminant Forms 156

4 Root Lattices 156

4.1 A;, A.. 157

4.2 D*/D, 157

4.3 F.„ 158

Contents xix

4.4 E^/Ev 158

4.5 E*/Es 158

5 Elliptic Fibrations 159

5.1 K3 Surfaces and Elliptic Fibrations 159

5.2 Nikulin and Niemeier's Results 161

5.3 Nishiyama's Method 162

6 Elliptic Fibrations of Y2 164

6.1 The Primitive Embeddings of D$ © A \ into Root

Lattices 164

6.2 Generators of L/Lroot 168

6.3 Lmot = ExDl(, 168

6.4 /-root 170

6.5 /-root = ^;v 170

6.6 Lmot = Dl 171

6.7 / mot-/V^ i s 171

6.8 /.root-/:,! 171

6.9 Lr0ol = AuEbD7 171

6.10 /,•«„,, - Dt 172

6.11 W= £M5 172

6.12 Lrooi = D]A] 173

7 Equations of the Fibrations 173

7.1 Equation of the Modular Surface Associated

to the Modular Group T, (8) 174

7.2 Construction of the Graph from the Modular Fibration..

175

7.3 Two Fibrations 176

7.4 Divisors 177

8 Fibrations from the Modular Fibration 179

9 A Second Set of Fibrations: Gluing and Breaking 186

9.1 Classical Examples 186

9.2 Fibration with a Singular Fiber of Type /„, n Large 187

9.3 Fibrations with Singular Fibers of Type /* 190

9.4 Breaking 192

10 Last Set 192

10.1 From Fibration of Parameter p 193

10.2 From Fibration of Parameter 8 195

References 198

Universal Kummer Families Over Shimura Curves 201

Amnon Besser and Ron Livne

1 Introduction 201

2 K3 Surfaces with Picard Number 19 and Twists of EllipticSurfaces 204

2.1 Lattices 204

2.2 Elliptic Surfaces 205

2.3 Quadratic Twists 206

2.4 The Basic Construction 207

2.5 The Neron-Severi and the Transcendental Lattices 208

XX Contents

3 The Moduli Map for Discriminants 6 and 15 211

3.1 Marked Elliptic Fibrations and Moduli Spaces 212

3.2 Types of Marked K3 Surfaces 213

3.3 Abelian Surfaces with Quaternionic Multiplication 216

3.4 The Associated Kummer Surface 218

3.5 The Basic Isomorphism 220

3.6 Local Monodromies 221

3.7 Case Number 1 on the List 223


4 Isogenics Between Abelian Surfaces and Discriminant Forms....

226

4.1 The Theory of Discriminant Forms 226

4.2 Rank 4 Lattices and Discriminant Forms 228

4.3 Applications to Isogenies of Abelian Varieties 231

4.4 Abelian Varieties with Multiplication by Eichler

Orders 233

4.5 Further Analysis 234

5 Isogenies Related to Abelian Surfaces with Quaternionic

Multiplication 235

5.1 A Special Subgroup 236

5.2 The Integral Cohomology of a QM Abelian Surface 236

5.3 The Type of the Special Subgroup 237

5.4 Discriminant Forms Associated with QM Abelian

Surfaces 238

6 A Special Isogeny 238

6.1 The Isogeny 238

6.2 A Converse Theorem 240

6.3 Level Structures 241

7 Isogenies and the Morrison Correspondence 242

7.1 The Morrison Correspondence 242

7.2 Nikulin Markings 243

7.3 The Neron-Severi Lattice of the Quotient Surface 244

7.4 The Precise Correspondence 245

7.5 A Transcendental Description 250

8 Explicit Computations 252

8.1 Number 5 on the List 253








Appendix 260

A. 1 Rational Invariants of Quadratic Forms Associated with SingularFibers 260

A. 1.1 Quadratic Forms Over Q/( 260

Contents xxi

A. 1.2 Quadratic Forms of Singular Fibers 261

A. 1.3 Ternary Forms of Quaternion Algebras 263

References 264

Numerical Trivial Automorphisms of Enriques Surfaces in ArbitraryCharacteristic 267

Igor V. Dolgachev

1 Introduction 267

2 Generalities 268

3 Lefschetz Fixed-Point Formula 270

4 Cohomologically Trivial Automorphisms 271

5 Numerically Trivial Automorphisms 275

6 Examples 276

7 Extra Special Enriques Surfaces 281

References 283

Picard-Fuchs Equations of Special One-Parameter Families

of Invertible Polynomials 285

Swantje Gahrs

1 Introduction 285

2 Preliminaries on Invertible Polynomials 287

3 The Picard-Fuchs Equation for Invertible

Polynomials and Consequences 290

3.1 The GKZ System for Invertible Polynomials 291

3.2 The Picard-Fuchs Equation 296

3.3 Statements on the Cohomology of the Solution Space ...299

3.4 The Case of Arnold's Strange Duality 301

3.5 Relations to the Poincare Series and Monodromy 305

References 309

A Structure Theorem for Fibrations on Delsarte Surfaces 311

Bas Heijne and Remke Kloosterman

1 Introduction 311

2 Delsarte Surfaces 313

3 Isotrivial Fibrations 323

References 332

Fourier-Mukai Partners and Polarised K3 Surfaces 333

K. Hulek and D. Ploog1 Review Fourier-Mukai Partners of K3 Surfaces 334

1.1 History: Derived Categories in Algebraic Geometry 334

1.2 Derived Categories as Invariants of Varieties 335

1.3 Fourier-Mukai Partners 336

1.4 Derived and Birational Equivalence 338

xxii Contents

2 Lattices 338

2.1 Gram Matrices 339

2.2 Genera 340

3 Overlattices 342

3.9 Overlattices from Primitive Embeddings 346

4 K3 Surfaces 350

5 Polarised K3 Surfaces 352

6 Polarisation and FM Partners 356

7 Counting FM Partners of Polarised K3 Surfaces

in Lattice Terms 358

8 Examples 361

References 364

On a Family of K3 Surfaces with -'/\ Symmetry 367

Dagan Karp, Jacob Lewis, Daniel Moore, Dmitri Skjorshammer,and Ursula Whitcher

1 Introduction 368

2 Toric Varieties and Semiample Hypersurfaces 369

2.1 Toric Varieties and Reflexive Polytopes 369

2.2 Semiample Hypersurfaces and the Residue Map 371

3 Three Symmetric Families of K3 Surfaces 374

3.1 Symplectic Group Actions on K3 Surfaces 374

3.2 An -7\ Symmetry of Polytopes and Hypersurfaces 374

4 Picard-Fuchs Equations 378

4.1 The Griffiths-Dwork Technique 378

4.2 A Picard-Fuchs Equation 379

5 Modularity and Its Geometric Meaning 381

5.1 Elliptic Fibrations on K3 Surfaces 382

5.2 Kummer and Shioda-lnose Structures Associated

to Products of Elliptic Curves 383

5.3 Modular Groups Associated to Our Families of K3

Surfaces 384

References 385

A ;nd of Elliptically Fibered K7> Surfaces: A Tale of Two Cycles 387

Matt Kerr

1 Introduction 387

2 Real and Transcendental Regulators 388

3 The Apery Family and an Inhomogeneous Picard-Fuchs

Equation 391

4 Af-Polari/ed K3 Surfaces and a Higher Green's Function 397

5 Proof of the Tauberian Lemma 2 405

References 408

Contents xxiii

A Note About Special Cycles on Moduli Spaces of K3 Surfaces 411

Stephen Kudla

1 Introduction 411

2 Special Cycles for Orthogonal Groups 412

2.1 Arithmetic Quotients 412

2.2 Special Cycles 412

3 Modular Generating Series 414

4 The Case of K3 Surfaces 416

4.1 Modular Interpretation of the Special Cycles 418

4.2 Some Applications 418

5 Kuga-Satake Abelian Varieties and Special Endomorphisms 423

5.1 The Kuga-Satake Construction 423

5.2 Special Cycles and Special Endomorphisms 425

References 426

Enriques Surfaces of Hutchinson-Gopel Type and Mathieu

Automorphisms 429

Shigeru Mukai and Hisanori Ohashi

1 Introduction 429

2 Rational Surfaces and Enriques Surfaces 432

3 Abelian Surfaces and Enriques Surfaces 434

4 Sextic Enriques Surfaces of Diagonal Type 439

5 Action of of Mathieu Type on Enriques Surfaces

of Hutchinson-Gopel Type 442

6 Examples of Mathieu Actions by Large Groups 446

7 The Characterization 451

References 453

Quartic K3 Surfaces and Cremona Transformations 455

Keiji Oguiso1 Introduction 455

2 Proof of Theorem 1(1 )(2) 456

3 Proof of Theorem 1(3) 458

References 460

Invariants of Regular Models of the Product of Two Elliptic Curves

at a Place of Multiplicative Reduction 461

Chad Schoen

1 Introduction 461

2 Notations 463

2.1 Basic Notations 463

2.2 Notations Related to the Closed Fiber, F, ol' n : £ -> T..

463

2.3 Notation Related to Components of V

and Their Intersections 464

3 The Weil Divisor Class Group of V 464

4 The Homology and Cohomology of V-j- 468

xxiv Contents

5 Variation of the Isomorphism Class of V 471

6 The Picard Group of V 473

7 Semi-stable Models and Small Resolutions 479

8 The Sheaves R'ftZ/n 481

8.1 Notations 481

References 487

Part III Research Articles: Arithmetic and Geometry of Calabi-Yau

Threefolds and Higher Dimentional Varieties

Dynamics of Special Points on Intermediate Jacobians 491

Xi Chen and James D. Lewis

1 Introduction 491

2 Some Preliminaries 492

3 Main Results 494

References 498

Calabi-Yau Conifold Expansions 499

Slawomir Cynk and Duco van Straten

1 Introduction 499

2 How to Compute Picard-Fuchs Operators 502

2.1 The Method of Griffiths-Dwork 502

2.2 Method of Period Expansion 503

3 Double Octics 506

4 An Algorithm 511

References 514

Quadratic Twists of Rigid Calabi-Yau Threefolds Over Q 517

Fernando Q. Gouvea, Ian Kiming, and Noriko Yui

1 Introduction 518

2 Quadratic Twists of Rigid Calabi-Yau Threefolds 518

2.1 Easy Examples of Twists 521

2.2 Self-fiber Products of Rational Elliptic Surfaces

with Section and Their Twists 522

2.3 The Schoen Quintic and Its Quadratic Twists 523

2.4 Explicit Description for a Holomorphic 3-Form

for a Complete Intersection Calabi-Yau Threefold 525

2.5 Two Rigid Calabi-Yau Threefolds of Werner

and van Geemen 525

2.6 The Rigid Calabi-Yau Threefold of van Geemen

and Nygaard 527

3 Remarks on the Levels of Twists 528

4 Final Remarks 528

4.1 An Explicit Unresolved Case 528

4.2 The Question About Existence of Geometric Twists 529

4.3 The Fixed Point Set of the Involution ( 530

References 532

Contents xxv

Counting Sheaves on Calabi-Yau and Abelian Threefolds 535

Martin G. Gulbrandsen

1 Virtual Counts 536

1.1 Deformation Invariance 536

1.2 Virtual Fundamental Class 536

1.3 Obstruction Theory 538

1.4 Behrend's Weighted Euler Characteristic 541

2 Abelian Threefolds 542

2.1 Determinants 542

2.2 Translation and Twist 544

References 547

The Segre Cubic and Borcherds Products 549

Shigeyuki Kondo

1 Introduction 549

2 The Segre Cubic Threefold 550

3 A Complex Ball Quotient 551

3.1 A Complex Ball 551

3.2 Roots and Reflections 553

3.3 Ball Quotient and Heegner Divisors 554

3.4 Interpretation via K2> Surfaces 555

4 Weil Representation 557

5 Borcherds Products 558

6 Gritsenko-Borcherds Liftings 561

References 564

Quasi-modular Forms Attached to Hodge Structures 567

Hossein Movasati

1 Introduction 567

2 Moduli of Polarized Hodge Structures 570

2.1 The Space of Polarized Lattices 570

2.2 Hodge Filtration 571

2.3 Period Domain U 572

2.4 An Algebraic Group 573

2.5 Griffiths Period Domain 574

3 Period Map 575

3.1 PoincareDual 575

3.2 Period Matrix 576

3.3 A Canonical Connection on £ 576

3.4 Some Functions on £ 577

4 Quasi-modular Forms Attached to Hodge Structures 578

4.1 Enhanced Projective Varieties 578

4.2 Period Map 580

4.3 Quasi-modular Forms 580

xxvi Contents

5 Examples 581

5.1 Siegel Quasi-modular Forms 582

5.2 Hodge Numbers. 1,1,1,1 584

References 586

The Zero Locus of the Infinitesimal Invariant 589

G. Pearlstein and Ch. Schnell

1 Introduction 589

2 Proof of the Theorem 591

2.1 Algebraic Description of the Zero Locus 591

2.2 A More Sophisticated Description 592

2.3 Zero Loci of Sections of Coherent Sheaves 595

3 Relation to Algebraic Cycles 598

3.1 Green-Griffiths Program 598

References 601

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