Lecture Notes in Mathematics Edited by A Dold and B. Eckmann

Series: Mathematisches lnstitut der Universitat Bonn Adviser: F. Hirzebruch

439

Kenji Ueno

Classification Theory of Algebraic Varieties and Compact Complex Spaces Notes written in collaboration with P. Cherenack

Springer-Verlag Berlin· Heidelberg· New York 1975

Dr. Kenji Ueno Department of Mathematics Faculty of Science University of Tokyo Tokyo/Japan

Library of Congress Cataloging in Publication Data

Ueno, Kenji, 1945-Classification theory of algebraic varieties and

compact complex spaces.

(Lecture notes in mathematics ; 439)

Bibliography: p.Includes index.

1. Algebraic varieties. 2. Complex manifolds3. Analytic spaces. 4. Fiber bundles (Mathematics)I. Title. II. Series: Lecture notes in mathematics(Berlin) ; 439 . QA3.L28 no. 439 [QA564] 510'.8s [514'.224] 75-1211

AMS Subject Classifications (1970): 14-02, 14A10, 14J15, 32-02,32C10, 32J15, 32J99, 32L05

ISBN 3-540-07138-5 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07138-5 Springer-Verlag New York · Heidelberg · Berlin

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© by Springer-Verlag Berlin · Heidelberg 1975. Printed in Germany.

Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

To Professor K. Kodaira

PREFACE

The present notes are based on the lectures which I gave at the

University of Mannheim from March 1972 to July 1972. The lectures

were informal and were intended to provide an introduction to the

classification theory of higher dimensional algebraic varieties and

compact complex spaces recently developed by s. Kawai, s. Iitaka and

other mathematicians in Tokyo. The notes were taken by P. Cherenack.

Since there were no available lecture notes on these subjects, I

decided after reading Cherenack's notes to rewrite them more systemat~

ically so that they would serve as an introduction to our classifica-

tion theory.

been added.

Several topics which I did not mention at Mannheim have

P. Cherenack typed a good part of the first version of my manu-

script, improving my English. He also compiled a first version of the

bibliography which was quite helpful in completing the final version of

the bibliography. Here I gratefully acknowledge my indebtedness to him.

I would like to express my thanks to Professor H. Popp and the

Department of Mathematics of the University of Mannheim for giving me

the opportunity of visiting Mannheim and of giving these lectures.

The greater part of the final version of the present notes was written

when I was a visiting member of the M:athematical Institute of the

University of Bonn. I wish to express my thanks to Professor

F. Hirzebruch and the Mathematical Institute of the University of Bonn

for inviting me to Bonn, and to the Department of Mathematics of the

University of Tokyo for giving me permission to visit Mannheim and Bonn.

My thanks are due to Dr. Y. Nami.kawa, Dr. E. Horikawa, Mr. T.

Fujita and Mr. Masahide Kato who read! the manuscript in whole or in

VI

part and pointed out some mistakes and suggested some improvements.

I wish to express my thanks to Professor H. Popp and Professor s.

Iitaka for their constant encouragement during the preparation of the

present notes. Last, but not least, I would like to express my thanks

to Miss K. Motoishi for her typewriting.

Tokyo, September 1974. Kenji Ueno.

Table of Contents

Introduction

Conventions and notations

Chapter I, Analytic spaces and algebraic varieties 1

§1, GAGA, Proper mapping theorem, Stein factorization, 3

§2. Meromorphic mappings and the resolution of singularities. • • • . • • . • • • • • . • • • • • • • • • • • • • • • . . . • • • . • . . 13

§3 Algebraic dimensions and algebraic reductions of complex varieties. . . • . . • • • • • • • • • • • • • • . • • • • • • • • • • . • . . 24

Chapter II, D-dimensions and Kodaira dimensions 28

§ 4. Divisors and linear systems ...• , • • • • • • • • • . . . . . • • . • • • • 30

§5.

§6.

D-dimensions and L-dimensions.

Kodaira dimension of a complex variety.

Chapter III, Fundamental theorems

Chapter IV

Proof of Theorem 5.10 and Theorem 6,11,

Asymptotic behaviour of j(mD).

Classification of algebraic varieties and complex varieties

§9, Albanese mappings and certain bimeromorphic

so

65

76

77

86

94

invariants. • • • • • • • • . • • • • • . . • • • • • • • • • • • • • • • • . • . . • • • . . 96

§10.

§11,

Chapter V,

Subvarieties of complex tori.

Classification theory. ..............................

Algebraic reductions of complex varieties and complex manifolds of algebraic dimension zero

116

125

141

§6.

§6.

VIII

§ 12. Algebraic reductions of complex varieties.

§ 13. Complex manifolds of algebraic dimension zero.

Chapter VI. Addition formula for Kodaira dimensions of analytic fibre bundles

§ 14. Pluricanonical representations of bimeromorphic

143

157

172

transformation groups •..........•.••.•...•..•.••••••. 173

§ 15. Addition formula

Chapter VII. Examples of complex manifolds

§16.

§ 17.

Kummer manifolds.

Complex parallelizable manifolds.

§18. Complex structures on a product of two odd-

185

188

190

212

dimensional homotopy spheres ••.•.•.••••••.•.•.•.••.. 228

Chapter VIII.

§19. Miscellaneous results. • •••••••••••.••••••••••••••••• 240

Appendix

§ 20. Classification of surfaces. .. ........................ 248

Bibliography 260

276 Appendix

INTRODUCTION

There are two notions of classification theory of complete

algebraic manifolds defined over ~ or compact complex manifolds,

a rough classification and a fine classification. For example, in the

case of non-singular curves, i.e. compact Riemann surfaces, we sub-

divide isomorphism classes of curves into infinitely many families

g=l, 2, 3, ... , where M g

consists of isomorphism classes of

curves of genus g. This gives a rough classification. The study of

the structure of the set M gives a fine classification of curves. g

This is usually called the theory of moduli. The study of the set

Mg is nothing but the study of all complex structures on a fixed

topological model of a compact Riemann surface of genus g. One of

the important results on the theory of moduli of curves is that M g

carries the structure of a quasi-projective variety (see Baily [1]

and Mumford [s]).

In these lecture notes we shall mainly discuss a rough classifi-

cation of birational (resp. bimeromorphic) equivalence classes of

complete algebraic manifolds (resp. compact complex manifolds).

But we shall show later that the theory of moduli, that is the finest

classification, is deeply related to a rough classification of compact

complex manifolds without non-constant meromorphic functions (see §13

below). On the other hand, since we are interested in birational

(resp. bimeromorphic) equivalence classes of algebraic manifolds (resp.

compact complex manifolds), by virtue of the resolution of singularities

due to Hironaka ([1], [2]), we can consider classification of algebraic

varieties and reduced, irreducible, compact complex spaces.

Thus one of the main purposes of our classification theory is to

X

find good birational (resp. bimeromorphic) invariants of algebraic

manifolds(resp. of compact complex manifolds), such as the genera of

curves so that we can subdivide birational (resp. bimeromorphic)

equivalence classes of manifolds into certain large families.

Let us recall briefly the classification theory of analytic sur-

faces, i.e. two-dimensional compact complex manifolds. The classifi-

cation of surfaces is much more complicated than the classification of

curves. Contrary to the case of curves, rwo bimeromorphically equiva-

lent surfaces are not necessarily analytically isomorphic to each other.

The difficulty is overcome using the theory of exceptional curves of

the first kind (see Definition 20.1 below and Zariski [4]). The theory

says that any analytic surface is obtained from a relatively minimal

model by finite succession of monoidal transformations. Moreover,

except for rational and ruled surfaces, two relatively minimal models

are bimeromorphically equivalent if and only if they are isomorphic to

each other (see Theorem 20. 3 ) • Thus it is enough to classify

relatively minimal models of surfaces.

Classification of algebraic surfaces was done partly by Castelnuovo

and mainly by Enriques. They found the important birational invariants,

the irregularity and the plurigenera of a surface, The irregularity

q(S) of a surface s is defined by

q(S) dimf:H1 (s, Q.g).

For a positive integer m, the m-genus p (S) m of a surface S

(if we do not specify the integer m, we call it a plurigenus of S )

is defined by

Pm(S) = dimf:H0(s, Q.(mK8)),

where K8 is the canonical bundle of the surface s. (Note that

the above definitions work for any compact complex manifold.)

According to whether P12 = 0, P12 = 1 or P12 ) 1, respectively,

they classified algebraic surfaces into three big classes (using the

Kodaira dimension K which will be defined in §6 below, we can say

that these three classes consist respectively of the algebraic surfaces

for which K = -~ K = 0, K > 0 ), and subdivided each class into

finer classes (see Enriques [ 1], p.463-464). Their arguments were

quite intuitive. Rigorous proofs were given by several mathematicians,

especially by Kodaira (see Kodaira [2], [3], Safarevic et al [1] and

Zariski [7], II, p.277-505).

Kodaira has generalized the classification theory of algebraic

surfaces to that of analytic surfaces (see Kodaira [2], [3]).

In Kodaira [ 2], I, we can find two important results on non-algebraic

analytic surfaces. The one is the algebraic reduction of analytic

surfaces and the other is the structure theorem of Kahler surfaces

without non-constant meromorphic functions. Let t(S) be the field

consisting of all meromorphic functions on an analytic surface s.

The transcendence degree over ~ of the function field ~(S) is called

the algebraic dimension a(S) of s. Kodaira has shown that if a(S)

=1, there exist a non-singular curve C and a subjective morphism ~ : S

---t C, which we call the algebraic reduction of S, such that ~

induces an isomorphism between the function fields ~(C) and ~(S),

and that general fibres of the morphism ~ are elliptic curves.

Kodaira has also shown that if a Kahler surface S is of algebraic

dimension zero, then the irregularity q(S) of S is zero or two, and

that if moreover q(S) = 2 the natural mapping of S into its Albanese

variety is a modification of s. These two results have been general-

XII

ized by Kawai in the case of three-dimensional compact complex manifolds

(see Kawai [1] and § 12, § 13 below). Kawai's results were the first

definite results on classification theory of higher dimensional compact

complex manifolds.

By the way, we have already mentioned that the pulrigenera play

an important role in the classification of surfaces. The plurigenera

are deeply related to the pluricanonical mappings. Let us suppose that

P (S) ~ 0 for an analytic surface S and a positive integer m. m

be a basis of the vector space

We define a meromorphic mapping P mK S---} lPN by

S PN w w z -----7 (t{'

0(z): 'f1 (z):···: <fN(z))

and call it the m-th canonical mapping. If the integer m is not speci-

fied, we call it a pluricanonical mapping. Enriques had already studied

the pluricanonical mappings of certain algebraic surfaces of general

type (see Enriques [l])and Kodaira has given a general theory of the

pluricanonical mappings of algebraic surfaces of general type. The

nature of the pluricanonical mappings of elliptic surfaces can be easily

deduced from the canonical bundle formula for elliptic surfaces due to

Kodaira (see Kodaira [3], I and the formula 20.13.1 below).

Inspired by these results, Iitaka has studied the pluricanonical

mappings of higher dimensional compact complex manifolds (see Iitaka

[2] and ~5- ~8 below). More generally he has studied meromorphic

mappings associated with complete linear systems of Cartier divisors

on a normal variety. He has defined the Kodaira dimensions of compact

complex manifolds and has proved a fundamental theorem on the pluri-

canonical fibrations (see Theorem 6.11 below). The Kodaira dimension

Appendix

XIII

K(M) of a compact complex manifold M is, by definition, - oo if

P (M) = 0 for every positive integer, and is the maximal dimension of m

the image varieties of M under the pluricanonical mappings if p (M) m

~ 1 for at least one positive integer m. The Kodaira dimension is

a bimeromorphic invariant of a given compact complex manifold. The

fundamental theorem on the pluricanonical fibrations due to Iitaka says

that if the Kodaira dimension K(M) of a compact complex manifold M

* is positive, then there exists a bimeromorphically equivalent model M

of M which has the structure of a fibre space whose general fibres are of

Kodaira dimension zero (for the precise statement of the theorem, see

Theorem 6 .11) •

As we have mentioned above, with each of the bimeromorphic

invariants, that is, with the algebraic dimension, with the Kodaira

dimension, with each of the plurigenera, and with the irregularity we

can associate a meromorphic mapping and introduce a fibre space struc-

ture on the compact complex manifold. With the algebraic dimension,

we can associate an algebraic reduction. With the Kodaira dimension

and with each plurigenus we can associate a pluricanonical mapping.

With the irregularity (if a complex manifold is neither algebraic nor

Kahler, we should replace the irregularity by the Albanese dimension),

we can associate the Albanese mapping. Using the fibre spaces introduced

by these mappings, we shall show that the classification theory is

reduced to the study of these fibre spaces and the study of special

manifolds.

In Iitaka [3], relying on the fundamental theorem of the pluri-

canonical fibrations and Kawai's results mentioned above, Iitaka has

discussed the classification theory of algebraic varieties and compact

XIV

complex spaces. After his paper a number of interesting results on

classification theory have been obtained. Nakamura has shown that the

Kodaira dimension is not necessarily invariant under small deformations

(see Nakamura [1] and § 17 below). Nakamura and Ueno have shown the

addition formula for Kodaira dimensions of analytic fibre bundles whose

fibres are Moishezon manifolds (see Nakamura and Ueno [1], and §15

below). This formula gives an affirmative answer to a special case of

Conjecture C (see ~ 11). n Ueno has studied Albanese mappings and has

shown that Albanese mappings play an important role in classification

theory (see Ueno [3] and Chapter IV below). In Ueno [3],the author

has also proved the canonical bundle formula for certain elliptic three-

folds and has studied Kummer manifolds (see §11 and ~16 below).

Akao has studied prehomogeneous Kahler manifolds (see Akao [2] and ~19).

Kato has studied complex structures on s1 x s5 (see Masahide Kato [2],

[3] and S 18). Iitaka has introduced new birational invariants and

has studied three-dimensional rational manifolds (see Iitaka [4] and

.r 19). He has also studied three-dimensional algebraic manifold whose

universal covering is the three-dimensional complex affine space (see

Iitaka [s]).

The main purpose of the present lecture notes is to provide a

systematic treatment of these subjects. Many examples of complex

manifolds exhibiting a difference between the classification of sur-

faces and that of higher dimensional complex manifolds will be given

(see Chapter VII below). At the moment our classification theory is

far from satisfactory but we already have a lot of interesting results.

We hope that the present notes will serve as an introduction to this

new field.

XV

For a complete understanding of these lecture notes,a knowledge of

the general theory of complex manifolds and of the classification theory

of surfaces is indispensable. On these subjects we refer the reader

to Kodaira and Morrow [1] and to Kodaira [2], [3]. Ueno [4], [s] will

serve as an introduction to the present notes.

Conventions and Notations

Unless otherwise explicitly mentioned, the following conventions

will be in force throughout these notes.

1) By an algebraic variety we mean a complete irreducible algebraic

variety defined over t. By an algebraic manifold we mean a non-

singular algebraic variety.

2) All complex manifolds are assumed to be compact and connected.

3) By a complex variety we mean a compact irreducible reduced complex

space.

4) The word "manifold" (resp. "variety") means an algebraic manifold.

(resp. algebraic variety) or a complex manifold (resp. complex variety).

5) For a Cartier divisor D on a variety D, by [D] we denote the

complex line bundle associated with the divisor D. We often identify

the sheaf Qv(mD) with the sheaf Q([mD]) by a natural isomorphism.

In §7 we shall distinguish these two sheaves.

6) For a complex line bundle L we often write mL instead of L@m.

7) By an elliptic bundle over a manifold M we mean a fibre bundle

over M whose fibre and structure group are an elliptic curve E and

the automorphism group Aut(E) of E, respectively.

8) The dimension of a variety is a complex dimension.

9) By a fibre space f : M--7 W of complex manifolds we mean that

a morphism f is surjective and that any fibre of f is connected.

10) For a Cartier divisor D on a complex variety V, we define

.Q.(mD) by

J.(mD)

where t v* ~ V is the normalization of V.

Appendix

a(V) ;

(A(M), a)

Aut(V) ;

Aut0 (V) ;

Bim(V)

O::(V)

gk (V) ;

I<M = K(M)

p (V) g

p (V) ; m

q(V) ;

r(V)

t (V)

a : M ---7 A(M)

K (D , V) ;

K (V)

XVIII

the algebraic dimensions of a complex variety V (see Definition 3,2).

the Albanese torus of a complex manifold M (see Definition 9,6).

the automorphism group of a complex variety v

the identity component of Aut(V).

the bimeromorphic transformation group of a complex variety V.

the meromorphic function field of a variety v

see Definition 9,20.

the canonical line bundle (a canonical divisor) of a manifold M,

the projective fibre space associated with a coherent sheaf K(see 2,8).

the geometric genus of a variety V(see Defi­nition 6,5).

the m-genus of a variety V (see Definition 6.5).

the irregularity of a variety V (see Defini­tion 9,20),

the k-th irregularity of a variety V (see Definition 9,20).

see Definition 9,20.

the Albanese dimension of a variety V (see Definition 9,21).

the Albanese mapping (see Definition 9,6).

the D-dimension of a variety V (see Defini­tion 5,1),

the Kodaira dimension of a variety V (see Definition 6,5).

the meromorphic mapping associated with a complete linear system lmDI (see 2.4 and ~5). If D is the canonical line bundle, we call it the m-th canonical mapping,

Appendix

Appendix

Appendix Appendix Appendix

XIX

a real k-dimensional homotopy sphere which bounds a parallelizable manifold.

the sheaf of germs of holomorphic k-forms on a complex manifold M

Appendix

Appendix