Complex Abelian Varieties

442 Pages·1992·8.71 MB·English

by  Herbert Lange

Grundlehren der
mathematischen Wissenschaften 302
A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S. S. Chern J. Coates J. M. Frohlich
H. Hironaka F. Hirzebruch L. Hormander
C. C. Moore J. K. Moser M. Nagata W Schmidt
D. S. Scott Ya. G. Sinai J. Tits M. Waldschmidt
S. Watanabe
Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Herbert Lange Christina Birkenhake
Complex
Abelian Varieties
Springer-Verlag Berlin Heidelberg GmbH
Herbert Lange
Christina Birkenhake
Mathematisches Institut
Universität Erlangen-Nümberg
Bismarckstr. 1 1/2
W-8520 Erlangen, FRG
Mathematics Subject Classification (1980): 14-02, 14KXX, 30FlO,
32G20
ISBN 978-3-662-02790-5 ISBN 978-3-662-02788-2 (eBook)
DOI 10.1007/978-3-662-02788-2
Library of Congress Cataloging-in-Publication Data
Lange, H. (Herbert), 1943-Complex Abelian varieties / Herbert Lange, Christina
Birkenhake. p. cm. --(Grundlehren der mathematischen Wissenschaften; 302)
Includes bibliographical references and index.
I. Abelian Varieties. 2. Riemann surfaces. I. Birkenhake, Christina. 11. Title.
III. Series. QA564.L32 1992 516.3'53--dc20 92-23806
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© Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992
Softcover reprint of the hardcover 1st edition 1992
Typesetting: Camera ready by authors
41/3140-543210 -Printed on acid-free paper
Contents
Introduction 1
Notation 5
Chapter 1
Complex Tori 6
§ 1 Complex Tori 7
§ 2 Homomorphisms 9
§ 3 Cohomology of Complex Tori 13
§ 4 The Hodge Decomposition 15
Exercises 20
Chapter 2
Line Bundles on Complex Tori 23
§ 1 Line Bundles on Complex Tori 24
§ 2 The Appell-Humbert Theorem 29
§ 3 Canonical Factors 32
§ 4 The Dual Complex Torus 35
§ 5 The Poincare Bundle 39
Exercises 42
Chapter 3
Cohomology of Line Bundles 46
§ 1 Characteristics 47
§ 2 Theta Functions 51
§ 3 The Positive Semidefinite Case 57
§ 4 The Vanishing Theorem 58
§ 5 Cohomology of Line Bundles 63
§ 6 The Riemann-Roch Theorem 66
Exercises 69
Chapter 4
Abelian Varieties 71
§ 1 Polarized Abelian Varieties 72
§ 2 The Riemann Relations 75
§ 3 The Decomposition Theorem 77
§ 4 The Gauss Map 82
§ 5 Projective Embeddings 85
§ 6 Symmetric Line Bundles 89
§ 7 Symmetric Divisors 93
§ 8 Kummer Varieties 98
§ 9 Morphisms into Abelian Varieties 101
§ 10 The Pontryagin Product 104
§ 11 Homological Versus Numerical Equivalence 107
Exercises 111
vi Contents
Chapter 5
Endomorphisms of Abelian Varieties 115
§ 1 The Rosati Involution 116
§ 2 Polarizations 121
§ 3 Norm-Endomorphisms and Symmetric Idempotents 125
§ 4 Endomorphisms Associated to Cycles 129
§ 5 The Endomorphism Algebra of a Simple Abelian Variety 133
Exercises 142
Chapter 6
Theta and Heisenberg Groups 147
§ 1 Theta Groups 148
§ 2 Theta Groups under Homomorphisms 152
§ 3 The Commutator Map 154
§ 4 The Canonical Representation of the Theta-Group 156
§ 5 The Isogeny Theorem 159
§ 6 Heisenberg Groups and Theta Structures 162
§ 7 The Schrodinger Representation 167
§ 8 The Isogeny Theorem for Finite Theta Functions 170
§ 9 Symmetric Theta Structures 173
Exercises 178
Chapter 7
Equations for Abelian Varieties 182
§ 1 The Multiplication Formula 183
§ 2 Surjectivity of the Multiplication Map 187
§ 3 Projective Normality 190
§ 4 The Ideal of an A belian Variety in IP N 194
§ 5 Riemann's Theta Relations 200
§ 6 Cubic Theta Relations 202
Exercises 208
Chapter 8
Moduli 212
§ 1 The Siegel Upper Half Space 213
§ 2 The Analytic Moduli Space 216
§ 3 Level Structures 220
§ 4 The Theta Transformation Formula, Preliminary Version 224
§ 5 Classical Theta Functions 227
§ 6 The Theta Transformation Formula, Final Version 231
§ 7 The Universal Family 233
§ 8 The Action of the Symplectic Group 236
§ 9 Orthogonal Level D-Structures 238
§ 10 The Embedding of AD(D)o into Projective Space 240
Exercises 244
Contents vii
Chapter 9
Moduli Spaces of Abelian Varieties with Endomorphism 247
Structure
§ 1 Abelian Varieties with Endomorphism Structure 249
§ 2 Abelian Varieties with Real Multiplication 250
§ 3 Some Notation 256
§ 4 Families of Abelian Varieties with Totally Indefinite 259
Quaternion Multiplication
§ 5 Families of Abelian Varieties with Totally Definite 262
Quaternion Multiplication
§ 6 Families of Abelian Varieties with Complex Multiplication 268
§ 7 Group Actions on 'Hr,. and 'Hm 273
§ 8 Shimura Varieties 276
§ 9 The Endomorphism Algebra of a General Member 280
Exercises 286
Chapter 10
Abelian Surfaces 288
§ 1 Preliminaries 289
§ 2 The 166-Configuration of the Kummer Surface 292
§ 3 An Equation for the Kummer Surface 297
§ 4 Reider's Theorem 301
§ 5 Polarizations of Type (1,4) 308
§ 6 Products of Elliptic Curves 313
Exercises 317
Chapter 11
Jacobian Varieties 320
§ 1 Definition of the Jacobian Variety 321
§ 2 The Theta Divisor 327
§ 3 The Poincare Bundles for a Curve C 333
§ 4 The Universal Property 336
§ 5 Correspondences 340
§ 6 Endomorphisms Associated to Curves and Divisors 342
§ 7 Examples of Jacobians 344
§ 8 The Criterion of Matsusaka-Ran 348
§ 9 Trisecants of the Kummer Variety 352
§ 10 Fay's Trisecant Identity 355
Exercises 361
Chapter 12
Prym Varieties 365
§ 1 Abelian Subvarieties of a Principally Polarized Abelian Variety 367
§ 2 Prym-Tyurin Varieties 371
§ 3 Prym Varieties 375
§ 4 Topological Construction of Prym Varieties 378
§ 5 The Abel-Prym Map 381
viii Contents
§ 6 The Theta Divisor of a Prym Variety 385
§ 7 Recilla's Theorem 390
§ 8 Donagi's Tetragonal Construction 393
§ 9 Kanev's Criterion 399
Exercises 405
Appendix
A Algebraic Varieties and Complex Analytic Spaces 409
B Line Bundles and Factors of Automorphy 412
Bibliograpy 417
Glossary of Notation 427
Index 431
Introdu ction
A hyperelliptic integral is by definition an integral of the form
J
dz
v'
'Y f(z) ,
where 'Y is a path in the complex plane C with coordinate z and J(z)
(z - a1) ... (z - ad) with pairwise different constants ai. If d = deg J is
1 or 2, an explicit integration by elementary functions is well known from
calculus. If d = 3 or 4, integration is possible using elliptic functions. If
however d 2 5, no explicit integration is known in general.
The reason for this is the following: the differential W = ~ is not
yf(z)
single valued, considered as a function on C. Let C denote the compact
.,fr
Riemann surface associated to By definition C is the double covering of
the Riemann sphere lP l' ramified at the points aI' ... , ad together with 00
if d is odd. Now w may be considered as a holomorphic differential on C. It
is essentially the topological structure of C which causes the problem. The
more complicated it is, the more difficult it is to integrate w.
At the beginning of the 19 th century the Norwegian mathematician Niels
Henrik Abel (1802-1829) and the German mathematician Carl Gustav Jacob
Jacobi (1804-1851) found a way to attack this problem. In geometric terms
their method can be described as follows. The idea is to try to integrate not
w alone, but simultaneously the whole set of holomorphic differentials
w.• = Zi-vl'fW~ £0 r z. = 1 , ... ,g= [-d2-l-]
on C. For this, fix a point Po E C and consider the map
p~ (Jw1, ... ,Jw
g),
Po Po
defined on a small neighbourhood U of Po. Unfortunately this map cannot
be extended to the whole of C, since the integrals depend on the path from
Po to p. However, if we consider it modulo the values of integrals along all
possible closed paths, we obtain a well-defined map. To be more precise, we
2 Introduction
consider HI (G, 1l) to be the group of closed paths starting from Po modulo
homology. The image of the map HI (G, 1l) --+ (C9 , 'Y 1-+ (II WI' ..• , II W 9) is
a lattice in (C9, i.e. a discrete subgroup ofrank 2g. So the quotient
is a complex torus, called the Jacobian variety of G. It can be shown
that J( G) is isomorphic to a projective variety. A complex torus with this
property is called an abelian variety. By construction the integration map
U --+ (C9 discussed above extends to a holomorphic map
(j j W9)
a: G --+ J(G) , p 1-+ WI"'" mod HI(G, 1l) ,
Po Po
called the Abel-Jacobi map. In these terms the integration of (WI"" ,W9) is
essentially equivalent to the following two steps:
(i) Determine the Jacobian variety J(G).
(ii) Describe the Abel-Jacobi map a: G --+ J( G).
Unfortunately only in very few cases can this be done explicitly (see Sec
tion 11.7). However, one can try to study the geometry of the pair (J (G), a),
and this may be considered as a sort of substitute for step (i). This method is
not restricted to hyperelliptic integrals, but works for holomorphic integrals
on any compact Riemann surface.
Of course Abel and Jacobi did not express their results in the above geo
metric language. For them integration was entirely a matter of analysis. They
worked with what we now call abelian functions, or rather abelian integrals.
Some essential progress in the subject was made by Riemann. Even if he
did not invent theta functions, he used them heavily in his investigations
of abelian functions. Apart from Riemann, the analytic theory of abelian
functions is mainly due to WeierstraB, Frobenius, Poincare and Picard. For
example, the fact that every abelian function is a quotient of theta functions
was known to WeierstraB in 1878 and proved by Poincare [1] in 1899.
Towards the end of the 19 th century geometers started to study the theory
of abelian and theta functions by geometric methods. Originally an "abelian
variety" of dimension 9 meant a hypersurface of IP 9+1 given as the image of
(C9 under the map defined by g+2 suitable theta functions (see de Franchis [1]
p. 53). Later this notion was extended to mean a projective variety given
as the image of (C9 under the map defined by any system of theta functions
of the same type (see Lefschetz [2] p. 355). However, since these varieties
often have unpleasant singularities and do not admit a group structure,
the language of complex tori turned out to be more fruitful for this purpose.
Picard seems to have been the first to express the theory of abelian functions
3
in this language. But it was only after the fundamental paper of Lefschetz [1]
that this point of view was generally accepted. The basics of the geometric
theory of abelian varieties are largely due to Scorza, Rosati, and Lefschetz.
Today abelian varieties play an important role in several branches of math
ematics. Number theory is certainly the subject in which they are most ex
tensively applied. They are heavily used in class field theory, as well as for
rationality and transcendence questions. In the theory of dynamical systems
they represent a tool to solve certain Hamiltonian systems. Their impor
tance in algebraic geometry lies in the fact that there are natural ways to
associate to any smooth projective algebraic variety V an abelian variety X
and investigate properties of V by studying X. Examples of this are the Pi
card variety, the Albanese variety, and certain intermediate Jacobians. Some
problems in physics can be studied via abelian varieties. Let us mention only
that theta functions are solutions of the heat equation in thermodynamics.
A more recent branch of physics applying abelian varieties is string theory.
Apart from their importance in applications, geometric properties of
abelian varieties are interesting for their own sake and this is the subject
of the present book. The central topics are the projective embeddings of
an abelian variety, their equations and geometric properties, discussed in
Chapters 7 and 10. Moreover we construct several moduli spaces of abelian
varieties with additional structure in Chapters 8 and 9 and give some appli
cations for the theory of algebraic curves in Chapters 11 and 12.
The main tool for the proofs is the theta group of a line bundle L on X, in
troduced in Chapter 6. Implicitly it appears already in Chapter 3, where we
construct a basis of canonical theta functions of HO (X, L) in such a way that
the theta group of L acts essentially by permutation (see Theorem 3.2.7).
Another tool is the notion of the characteristic of a nondegenerate line bun
dle on X. Characteristics represent a list of all line bundles in an algebraic
equivalence class; their definition is a direct generalization of the classical
notion of characteristics of theta functions. For more details concerning the
contents we refer to the introductions at the start of every chapter.
In order to be able to present some more advanced results, we restrict
ourselves to abelian varieties over the field <D of complex numbers. The main
advantage is that a line bundle on a complex torus <D9 / A can be described
by factors of automorphy on the universal covering <D9. In Appendix B we
give a short introduction and compile some results on factors of automorphy.
At the end of each chapter there is a section with exercises of varying
degrees of difficulty. Many of the exercises are standard results which were
not included in the text due to lack of space. Others contain theorems which
we quote from research papers and which are intended as hints for further
reading.