Complex Abelian Varieties and Theta Functions

104 Pages·1991·3.05 MB·English

by  George R. Kempf

George R. Kempf
Complex Abelian
Varieties and
Theta Functions
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
George R. Kempf
Department of Mathematics
John Hopkins University
Baltimore, MD 21218, USA
Mathematics Subject Classification (1980):
14K20, 14K25, 32C35, 32125, 32N05
ISBN 3-540-53168-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-53168-8 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-m-Publication Data
Kempf, George
Complcx abelian varieties and theta functions/George R Kcmpf p. cm -
(Universitext) Includes bibliographical references and index
ISBN 0-387-53168-8 - ISBN 3-540-53168-8
1. Abelian varieties. 2 Functions, Theta I Title
QA564 K45 1990
5163'53 - dc20 90-22573 CIP
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Preface
The study of abelian varieties began with the one-dimensional case of elliptic
curves. As such cill'ves are defined by a general cubic polynomial equation in
two variables, their study is basic to all but the simplest mathematic!:>. The
modern approach to elliptic curves occurred in the beginning of the nineteen
century with the work of Gauss, Abel and Jacobi.
Since the classical period there have been many developments in mathemat
ics. There are basically two distinct lines of generalization of an elliptic curve.
They are algebraic curves of higher genus' > 1. The other is higher dimensional
compact algebraic groups (abelian varieties).
This book deals with these higher dimensional objects which surprisingly
enough have more similar properties to elliptic curves than curves of higher
genus. There are three methods for studying abelian varieties: arithmetic, al
gebraic and analytic. The arithmetic study properly using both the algebraic
and analytic approaches and reduction modulo a prime. Mumford's book [3]
presents an adequate introduction to the algebraic approach with some indi
cation of the analytic theory. In this book I have restricted attention to the
analytic approach and I try to make full use of complex Hermitian geometry.
In this book I give the basic material on abelian varieties, their invertible
sheaves and sections, and cohomology and associated mappings to projective
spaces. I also provide an introduction to the moduli (parameter spaces for
abelian varieties) and modular functions. Lastly I give some examples where
abelian varieties occur in mathematics.
Some of the material is parallel to that found in Igusa's book [1], but I
have tried to develop the subject geometrically and avoid the connection with
representations of infinite non-abelian groups in Hilbert space. The book brings
some developments from the literature to book fonn; for example, Mumford's
theory of the theta group acting on the space of sections of invertible sheaves.
It seems an impossible tabk to give a proper bibliography and history of the
last two hundred years. As we desire, one generation's theorems have become
VI Preface
examples of the next generation's theories. For the people (some VLTj famous)
we don't mention explicitly who have participated in the historical development
of this branch of mathematics, we give thanks for their efforts.
Baltimore, August 1990 George R. K empj
Table of Contents
Chapter 1. Complex Tori
§ 1.1 The Definition of Complex Tori 1
§ 1.2 Hermitian Algebra ............................................. . 2
§ 1.3 The Invertible Sheaves on a Complex Torus .................... . 3
§ 1.4 The Structure of Pic(V/ L) ..................................... . 5
§ 1.5 Translating Invertible Sheaves ................................. . 7
Chapter 2. The Existence of Sections of Sheaves
§ 2.1 The Sections of Invertible Sheaves (Part I) ...................... 9
§ 2.2 The Sections of Invertible Sheaves (Part II) ..................... 10
§ 2.3 Abelian Varieties and Divisors .................................. 13
§ 2.4 Projective Embeddings of Abelian Varieties ..................... 15
Chapter 3. The Cohomology of Complex Tori
§ 3.1 The Cohomology of a Real Torus ........................ 19
§ 3.2 A Complex Torus as a Kahler Manifold ......................... 20
§ 3.3 The Proof of the Appel-Humbert Theorem...................... 21
§ 3.4 A Vanishing Theorem
for the Cohomology of Invertible Sheaves .................. . . . . .. 23
§ 3.5 The Final Determination
of the Cohomology of an Invertible Sheaf ....... . . . . . . . . . . . . .. .. 25
§ 3.6 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Chapter 4. Groups Acting on Complete Linear Systems
§ 4.1 Geometric Background ......................................... 29
§ 4.2 Representations of the Theta Group ............................ 31
rex,
§ 4.3 The Hermitian Structure on fi') ........................... 33
§ 4.4 The Isogeny Theorem up to a Constant ......................... 35
Chapter 5. Theta Functions
§ 5.1 Canonical Decompositions and Bases. . . . . ... .... ...... .. . . . . . ... 37
§ 5.2 The Theta Function ............................................ 38
VIII Table of Contents
§ 5.3 The lsogeny Theorem Absolutely ............................... 39
§ 5.4 The Classical Notation .......................................... 40
§ 5.5 The Length of the Theta Functions ............................. 42
Chapter 6. The Algebra of the Theta Functions
§ 6.1 The Addition Formula .......................................... 45
§ 6.2 Multiplication ........................................ '" . . ... . . 47
§ 6.3 Some Bilinear Relations ........................................ 49
§ 6.4 General Relations............... ............... ..... .. .. . . . . . . .. 51
Chapter 7. Moduli Spaces
§ 7.1 Complex Structures on a Symplectic Space. ... .... ... . .... ... .. . 55
§ 7.2 Siegel Upper-half Space.. . . . .. . .. . .. . . .. . . . . . . . . . . . . . .. .. . . . . . . . 58
§ 7.3 Families of Abelian Varieties and Moduli Spaces ................ 62
§ 7.4 Families of Ample Sheaves on a Variable Abelian Variety ........ 63
§ 7.5 Group Actions on the Families of Sheaves ....................... 66
Chapter 8. Modular Forms
§ 8.1 The Definition .................................................. 69
§ 8.2 The Relationship Between 7r~.KA and H
in the Principally Polarized Case ................................ 70
§ 8.3 Generators of the Relevant Discrete Groups .... ........... .... 72
§ 8.4 The Relationship Between 7r~.KA and H is General .............. 76
§ 8.5 Projective Embedding of Some Moduli Spaces .............. 77
Chapter 9. Mappings to Abelian Varieties
§ 9.1 Integration.. .... .. . . . . . . . . . . . . . . . . . . . .. . . . . . . ... .. . . . . . . . . . . . . . 81
§ 9.2 Complete Reducibility of Abelian Varieties ...................... 82
§ 9.3 The Characteristic Polynomial of an Endomorphism ............. 83
§ 9.4 The Gauss Mapping ............................................ 84
Chapter 10. The Linear System 12DI
§ 10.1 When IDI Has No Fixed Components........................... 87
§ 10.2 Projective Normality of 12DI .................................... 88
§ 10.3 The Factorization Theorem ..................................... 89
§ 10.4 The General Case .............................................. 90
§ 10.5 Projective Normality of 12DI on X/{±l} ........................ 92
Chapter 11. Abelian Varieties Occurring in Nature
§ 11.1 Hodge Structure ................................................ 95
§ 11.2 The Moduli of Polarized Hodge Structure ....................... 97
Table of Contents IX
§ 11.3 The Jacobian of a lliemann Surface ............................. 98
§ 11.4 Picard and Albanese Varieties for a Kahler Manifold ............ 99
Informal Discussions of Immediate Sources ...................... 101
References ........................................................... 103
Subject Index 105
Chapter 1. Complex Tori
§ 1.1 The Definition of Complex Tori
The lattice L in a real or complex finite dimensional vector space V is a discrete
subgroup such that the quotient group V / L is compact. The lattice L is a
free Abelian group of rank equal to the real dimension of V and the induced
®71
mapping L lR ~ V is an isomorphism and conversely.
A complex torus X = V / L is a complex vector space V modulo a latti.ce
L. Thus a complex torus is a commutative compact complex C-analytic group.
The complex tangent space Lie(X) of X at the identity 0 is naturally identi
fied with V. The quotient homomorphism V ~ V/L is just the exponential
mapping. exp: Lie(X) ~ X.
This explicit picture of complex tori is complemented by an :1bstract char
acterization (for those who know the clements of complex Lie group theory).
Theorem 1.1. Any compact connected C-analytic group X is a complex torus.
Proof. First we assume that X is known to be commutative. Then the expo
nential exp : Lie(X) ~ X will be a C-analytic homomorphism which is locally
an isomorphism. The last property implies that the kernel L of exp is discrete.
As X is connected exp is surjective and, hence, X ~ Lie(X)/ L. Thus L is a
lattice as X if:> compact.
To see that X is commutative consider the adjoint representation Ad : X ~
Aut(Lie(X)). This is a C-analytic mapping from a compact variety to an affine
variety. Thus Ad(X) = Identity. Hence To Ad = ad : Lie(X) ~ End(Lie(X))
is zero. In other words Lie(X) is an abelian Lie algebra. Consequently the
connected group X is commutative. 0
To get a rough idea of the possible complex tori V / L, we assume that
11, ... ,lg, m1, ... ,mg is a basis for L and that 11, ... ,Ig arc a complex basis
of C (we may always make such a choice). Hence 9 = dim(;(V). Then mj =
I:1::;;::;g (Xi,il; where ((Xi,;) is a 9 X 9 complex matrix. The condition that we
have a lattice is that 11, ... ,lg, mI, ... ,mg are IR-linearly independent or, what
2 Chapter 1. Complex Tori
is the same, Im( (Xi,;) is an invertible matrix. The abstract moral of this is that
the space of complex tori with a properly marked basis is naturally an analytic
manifold of diml (in this case the (Xi,; are global coordinates on these spaces).
Exercise 1. Let X be a complex torus of dimension g. Then for any non-zero
integer on multiplication by m : X ~ X is a surjective homomorphism with
finite kernel Xm and #Xm = Iml2g.
Exercise 2. Let I/> : V / L ~ V' / L' be a homomorphism of C-analytic groups
between two complex tori. Show that there is a unique C-linear mapping A :
V ~ V' such that A(L) eL' which induces 1/>.
Exercise 3. Let I/> : X ~ X' be a homomorphism of two complex tori of the
same dimension. Then I/> is surjective if and only if the kernel of I/> is finite.
A homomorphism satisfying the equivalent condition of Exercise 3 is called
an isogeny of degree = # kernel.
§ 1.2 Hermitian Algebra
Let V be a complex vector space. Recall that a Hermitian form H is a pairing
H: VxV ~ C such that H(z, w) is complex linear in z and H(z, w) = H(w, z).
It follows that H(z, w) is anti-complex linear in wand H(z, z) is a real-valued
quadratic form on V.
Let E(z,w) = ImH(z,w) be the imaginary part of a Hermitian form H.
Then
a) E is a real skew-symmetric form on Vas ImH(z,z) = 0 and
b) E(iz,iw) = E(z,w) as H(iz,iw) = i(-i)H(z,w) = H(z,w) we may
recover H from E.
Proposition 1.2. Given a form E satisfying a) and b) there is a unique Her
mitian form H with imaginary part E.
Proof. We check the uniqueness first. H(z, w) = ReH(z, w) + i 1m H(z, w) =
ImH(iz,w) +ilmH(z,w). Thus
H(z,w) = E(iz,w)+iE(z,w) .
This shows uniqueness. Conversely given E define H by the formula. Then we
need to check that H(iz, w) = iH(z, w) and H(w, z) = H(z, w). These two
equations follow easily from the properties a) and b). 0
§ 1.3. The Invertible Sheaves on a Complex Torus 3
We have a relationship between the properties of these forms H and E.
Recall that KerH = {z E VIH(z,w) = 0 for all w in V} and KerE = {z E
VIE(z,w) = 0 for all w in V}.
Lemma 1.3. Ker H = Ker E.
Proof. As E = 1m H, Ker E :J Ker H. Conversely by b) Ker E is invariant
under multiplication by i. Thus the formula (*) implies that Ker E ;2 Ker H.
o
Corollary 1.4. H is non· degenerate if and only if E is non-degenerate.
Exercise 1. Let H(z, z) = I:l<;<g ZjZj be the standard. Hermitian matrix
on C9. Write the associated Hermitian form H( z, w) and its imaginary part
E(z, w).
Exercise 2. In the situation of Lemma 1.3 assume that there is a lattice L in
V such that E is integral on Lx L. Then Ker(H) n L is a lattice in Ker H and
VI
the image L' of L in Ker H is a lattice. Furthermore H is indllced from a
VI
non-denegerate Hermitian H' on Ker H which is integral valued on L' X L'.
§ 1.3. The Invertible Sheaves on a Complex Torus
There is a general method for constructing a holomorphic line bundle L" on
the complex torus VI L by taking the quotient of a line bundle on V. Let L'
be a line bundle over V. Assume that we have a group action a : L X L' ~ L'
such that for any 1 in L and v in V Q(1, -) induces a linear isomorphism from
the fiber of L' over v to that over 1 + v. Then the quotient L" = L' I L is a line
VI
bundle over L.
In our situation we will consider the case where L' is trivial. Fix an isomor
phism B : C X V ~ L' of btmdles over V. Then T = B( 1, v) is an (arbitrary)
nowhere vanishing holomorphic section of L'. Then for any 1 in L
a(1, T( v)) = A,( v) . T(l + v)
where A, is a nowhere zero C-analytic function of v in V. The condition that
a is a group action is that a(1I, a (12 , T( v))) = a(11 + 12, T( v)) or
All (v + 12)A'a( V)T(11 + 12 + v) = a(1t, A'2(v )r(12 + v)) = A'l+12(V)T(11 + 12, v) .
Therefore we have the cocycle condition
c) All (v + 12 )A'2 (v) = A'd12 (v) for all 11, 12 in L and v in V.