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
Tills work is subject to copynght. All rights are reserved, whether the whole or part of the matenal is concerned, specifically the nghts of translation, repnnting, reuse 01 illustrations, recitation, broadcasting, reproduction on microfilms or in other \\ays, and storage m data banks Duplication of this publication or parts thereof is only permitted under the provisions of the German Copynght Law of September 9, 1965, in its current version, and a copynght fee must always be prud Violations fall under the prosecution act of the German Copyright Law © Springer-Verlag Berlin Heidelberg 1991 Printed m Germany 4113140-543210 - Pnnted on acid-free paper
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 mathematics. 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, algebraic 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 indication 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 § 1.2 § 1.3 § 1.4 § 1.5
The Definition of Complex Tori Hermitian Algebra ............................................. . The Invertible Sheaves on a Complex Torus .................... . The Structure of Pic(V/ L) ..................................... . Translating Invertible Sheaves ................................. .
1 2 3 5 7
Chapter 2. The Existence of Sections of Sheaves
§ 2.1 § 2.2 § 2.3 § 2.4
The Sections of Invertible Sheaves (Part I) ...................... The Sections of Invertible Sheaves (Part II) ..................... Abelian Varieties and Divisors .................................. Projective Embeddings of Abelian Varieties .....................
9 10 13 15
Chapter 3. The Cohomology of Complex Tori
§ 3.1 § 3.2 § 3.3 § 3.4 § 3.5 § 3.6
The Cohomology of a Real Torus ........................ A Complex Torus as a Kahler Manifold ......................... The Proof of the Appel-Humbert Theorem...................... A Vanishing Theorem for the Cohomology of Invertible Sheaves .................. . . . . .. The Final Determination of the Cohomology of an Invertible Sheaf ....... . . . . . . . . . . . . .. .. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19 20 21 23 25 26
Chapter 4. Groups Acting on Complete Linear Systems
§ 4.1 § 4.2 § 4.3 § 4.4
Geometric Background ......................................... Representations of the Theta Group ............................ The Hermitian Structure on rex, fi') ........................... The Isogeny Theorem up to a Constant .........................
29 31 33 35
Chapter 5. Theta Functions
§ 5.1 § 5.2
Canonical Decompositions and Bases. . . . . ... .... ...... .. . . . . . ... The Theta Function ............................................
37 38
VIII
Table of Contents
§ 5.3 § 5.4 § 5.5
The lsogeny Theorem Absolutely ............................... The Classical Notation .......................................... The Length of the Theta Functions .............................
39 40 42
Chapter 6. The Algebra of the Theta Functions
§ 6.1 § 6.2 § 6.3 § 6.4
The Addition Formula .......................................... Multiplication ........................................ '" . . ... . . Some Bilinear Relations ........................................ General Relations............... ............... ..... .. .. . . . . . . ..
45 47 49 51
Chapter 7. Moduli Spaces
§ 7.1 § 7.2 § 7.3 § 7.4 § 7.5
Complex Structures on a Symplectic Space. ... .... ... . .... ... .. . Siegel Upper-half Space.. . . . .. . .. . .. . . .. . . . . . . . . . . . . . .. .. . . . . . . . Families of Abelian Varieties and Moduli Spaces ................ Families of Ample Sheaves on a Variable Abelian Variety ........ Group Actions on the Families of Sheaves .......................
55 58 62 63 66
Chapter 8. Modular Forms
§ 8.1 § 8.2 § 8.3 § 8.4 § 8.5
The Definition .................................................. The Relationship Between 7r~.KA and H in the Principally Polarized Case ................................ Generators of the Relevant Discrete Groups .... ........... .... The Relationship Between 7r~.KA and H is General .............. Projective Embedding of Some Moduli Spaces ..............
69 70 72 76 77
Chapter 9. Mappings to Abelian Varieties
§ 9.1 § 9.2 § 9.3 § 9.4
Integration.. .... .. . . . . . . . . . . . . . . . . . . . .. . . . . . . ... .. . . . . . . . . . . . . . Complete Reducibility of Abelian Varieties ...................... The Characteristic Polynomial of an Endomorphism ............. The Gauss Mapping ............................................
81 82 83 84
Chapter 10. The Linear System 12DI § 10.1 § 10.2 § 10.3 § 10.4 § 10.5
When IDI Has No Fixed Components........................... Projective Normality of 12DI .................................... The Factorization Theorem ..................................... The General Case .............................................. Projective Normality of 12DI on X/{±l} ........................
87 88 89 90 92
Chapter 11. Abelian Varieties Occurring in Nature
§ 11.1 Hodge Structure ................................................ § 11.2 The Moduli of Polarized Hodge Structure .......................
95 97
Table of Contents
§ 11.3 The Jacobian of a lliemann Surface ............................. § 11.4 Picard and Albanese Varieties for a Kahler Manifold ............
IX
98 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 mapping L ®71 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 identified 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 characterization (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 exponential 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 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 11, ... ,lg , m1, ... ,mg
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 1. Determine explicitly all A.-H data for V / L. What is the degree of
L(a,H)? Exercise 2. In general show that
Exercise 3. As above, if we change the basic section r( v) of L' to r( v) B( v) where B (v) is as nowhere zero holomorphic function on V. Then the factor of automorphy which describes the same bundle L" would change to the cohomologous cocycle A,(v) B(l + v) B(l)-l.
91.4
The Structure of Pic (V/ L)
5
§ 1.4 The Structure of Pic(V/L) The Picard group Pic(V/ L) of a complex torus V / L is the group of isomorphism classes of invertible sheaves on V / L with tensor product as group law. By the last section we know that Pic(V/ L) = {A.-H. data for L in V}.
Lemma 1.6. Given an Hermitian form H on V such that the imaginary part E is integral on L X L we have exactly 22 dime va: L ~ {±l} such that (a, H) are A.-H. data.
Proof. Let 11, . .. ,12g be a basis for the lattice L where g = dim V. Given signs a(l;) in {±l} for each i, let a(En;I;) = [Ia(li)n;(_1)L:;<jn i nj E(I;,ld. Then one simply checks to (a, H) are exactly the required A.-H. data. 0 Let Pico = Hom71(L, U(l». Then we may regard Pico as a subgrollp of Pic(V/ L). Here is a homomorphism a: L ~ U(l) correspondence to the A.-H. data (a,O). Thus by Lemma 1.6 we have an exact sequence 1 ~ Pico ~ Pic(V/L) ~ {H sllch that ImH(L xL) C Zl} ~ 0. (By construction Pico corresponds to the unitary flat invertible sheaves on V/L.) Clearly Pic D is a real torus. We will next define a complex stlUcture on Pico. As a real torus we have the exact sequence
°
2'1fi-
~ Homz(L, Zl) ~ Hom71(L, ill) e~ HomL(L, U(l» ~ 0.
Thus we want to put a complex structure on the real vector space Hom71(L, R); i.e. define i).. for any homomorphism).. : L ~ IR. Extend)" to a real linear functional>. on V. We may do this uniquely because L is a lattice in V. Define i)..( I) == - 5.( il). Thus Pico = Homm.(V, R)/ Hom71( L, Zl) is a complex torus dual to V / L. We will need to test the appropriateness of this complex structure of Pico. If J-L : L ~ C* is a homomorphism, then J-L( I) are constant factors of automorphy for a flat invertible sheaf M(J-L) on V / L. We will check that M(J-L) corresponds to a point m(J-L) of Pico. Thus we will have a natural mapping m: Hom71(L,C*) ~ Pico. We intend to check.
Lemma 1.7. m is a complex analytic homomorphism for the obvious complex group structure on Hom71( L, C*).
Proof. Write J-L(l) = e21rilc (l) where k : L ~ C is a homomorphism. Let k : V ~ C be the real expansion of k. We have a complex linear mapping p(k) = p :
6
Chapter 1. Complex Tori
v ~ C given by p(v) = Imk(iv) +iImk(v). Thus R(k) = k - p is real valued. Now p(l)e- 211"i p(v+I)(e- 211"ip(v»)-1 = e211"i(k-p)(l) = e211"iR(k)(l) is cohomologous to p(l) and has values in U(l). Thus m(p)(l) = e2 11"iR(k)(I). We want to prove R: HomlR.(V, C) ~ HomlR.(V, IR) is a complex linear mapping. Clearly it is real linear. For the rest we want R( ik)( v) = - R(k)( iv) but this is readily verified as R(k)(v) = Re k(v) - Imk(iv). 0 There is an interesting invertible sheaf on the product complex torus X Pico(VIL) which is called the Poincare sheaf~. It has the nice property that ~IV/LX{(Ct,O)} is isomorphic to the sheaf .!l'(a,O) on VIL. Consider the Hermitian form H((u,u'),(v,v')) = -u'(iv) - v'(iu) + i(u'(v) - v'(u)) where u and v are in V and u' and v' are in HomlR.(V,IR) = V~. The lattice L~ in V~ which gives Pico is P E V~IA(L) ~ Zl}. The imaginary part E of H is +u'(v) - v'(u) and, hence, is integral on L x L~. Let B : L~ ~ tf/(1) be B(l, 1') = (-1)"(1). It is easy to check that (B, H) is A.-H. data for a sheaf ~ on VIL x Pico(VIL).
VIL
Exercise 1. Check that the Poincare sheaf has the above nice property. Exercise 2. Let (a(€),H(€)) be a continuous family of A.-H. data on VIL. Show that it has the form ((J(t)-y, k) where (J(t) is a continuous function with values in Pico and Cr, k) is a constant A.-H. data. Exercise 3. Let (a, H) be A.-H. data. Then a(O) = 1 and a(l) = a( _1)-1. Exercise 4. Show that the (a, H)'s of Lemma 1.6 are exactly those A.-H. data such that (-l)*(.!l'(a,H)) ~ .!l'(a,H) where -1 : VIL ~ VIL is the inverse morphism. (These are called the symmetric sheaves as VI L.) Exercise 5. Show that the double dual Pico of Pic° of VILis canonically isomorphic to VI L. Exercise 6. Show that dim Pico = dim VI L. Exercise 7. Let f : VI L have a natural isogeny f~
~
V' I L' be an isogeny of complex tori. Then we
: PicO(V' I L')
~
Pico(VI L)
and Ker(J~) is canonically the character group Hom(Ker(J), U(l)) of Ker(J). Conclude that degf~ = degf .
§ 1.5 Translating Invertible Sheaves
7
§ 1.5 Translating Invertible Sheaves In this section we will consider how invertible sheaves change Wlder the action of the group theoretic operations in a complex torus VI L. The idea is to take advantage of the fact that the exponents of the factors of automorphy are linear in v and quadratic in I. This is a much simpler situation than one might imagine without knowledge of the Appel-Humbert Theorem 1.5. Let x be a point of X = VI L. Then Tx : X ~ X denotes translation by x; i.e. Tx(Y) = x + y. Let !l' be an invertible sheaf on X. Consider the sheaf T;!l' 0 !l'0- 1 011 X. This is a continuous fWlction of x and is trivial when x = O. Thus one expects that there is a point
!e
L
Pico(y) .
Chapter 2. The Existence of Sections of Sheaves
§ 2.1 The Sections of Invertible Sheaves (Part I) Let (a,H) be A.-H. data for a complex torus V/L. Our objective is Theorem 2.1. The space r(v/ L,!l'( a, H)) of holomorphic sections of !l'( a, H) is non-zero if and only if both a) H is positive semi-definite and b) a is identically one on L n Ker H. The proof will be given in two sections. Here we will reduce to the most interesting case where H is positive definite.
Step 1. r( V / L, !l'( a, H)) = 0 ifthere exists v in V such that H ( v, v) < O. The best idea here is to appeal implicitly to the curvative property of the natural Hermitian metric in !l'( a, H). This metric is interesting for other reasons. To define it let f( v) = c- 1rH (v,tI). Then f is positive IR-analytic function on V and it satisfies the functional equation f(v
+ I) = e- 21rRe H(v,/)-1rH(/,/) f(v)
for alII and v in V. Let a then HI = a.
Let w
=
jdzI be an element of HI
= Ker D. Then
Thus 'L. iEl hi < a if w =1= a. Thus we are done if hk is bigger than 'L.iEl-k hi. VIle may make a diagonal change of coordinates such that this is true and not change the cohomology upto isomorphism. Thus HI = a. 0
Corollary 3.8. Hi(X,fi') = EBN~/~S'N HI . #1=;
§ 3.5 The Final Determination of the Cohomology of an Invertible Sheaf Let (a, H) be A.-H. data for an invertible sheaf fi' on a complex torus X = V / L. Let z = dim Ker H. Let Y = V/Ker(H) + L = (V/KerH)/L'. Then H is induced from a Hermitian definite form H' on V / Ker H. Let n be the number of negati ve eigenvalues of H. Let KO (fi') = Ker H / L n Ker H as usual. Then we have our objective
Theorem 3.9. a) Hi(X, fi') = a if i < n or i > n + z. b) If a ~ i ~ z, Hn+i(x, fi') ~ Hn(x, fi') 0 Hi (KO(fi') , 19K O(!C»). c) If alLnKer H ¢. 1, then Hn(x, fi') = a and, otherwise, dim Hn(x, fi') = JdetL' E' where E' = .~imH'. Proof. Part a) follows directly from Corollary 3.8. We may choose our coordinates such that hi = a if 1 ~ i ~ z and hi < a if z + 1 ~ i ~ z + n. To
26
Chapter 3. The Cohomology of Complex Tori
prove b) we will show that for any J C [1, ... ,zl multiplication by dZ J gives an isomorphism H[Z+l, .. ,z+nj with HJU[z+l, ... ,z+nj' This will prove b) by Corollary 3.8 and Theorem 3.3. On the other hand the mapping is an isomorphism by the differential equation of the HI as hi = 0 if 1 ::; i ::; z. For c) we introduce a new complex structure on the real vector space V = (;g. In the new complex structure Zj, ... ,Zz, Zz+l, . .. ,zz+n, Zz+n+l' ... ,Zg are the complex coordinates. Let V be this new complex space. Let H(z, w) be the Hermitian fonn EZ+l< 'z+n hi ZjWj.~Then H ~and _1_ ~ H have the same imaginary part. Now (a, H) is A.-H. data for V / Land H is semi-positive. We claim
If we prove this claim then part c) will follow from Theorem 2.1, its proof and Theorem 2.3 where we determined the sections of sheaves. Now for the claim let N = [z+l, ... ,z+nl. Thus Hn(x,fi') = HN and its elements correspond to forms W = fdzN in AOdz such that Llw = O. By the usual reasoning in Kiihler geometry this differential equation is equivalent to 8w = (8)*w = 0 ¢:} =0
H. )
if j
rt Nand (8~')* f )
= 0 (or, rather, - 8~')
f + 7rh f j
= 0) if j EN.
The isomorphism Hn(x,fi') ~ HO(V /L,fi'(a,H)) will send W = f dZN to g( z) = exp( -7r EiEN h j Zj Zj )f( z). The differential equations for 9 are = 0 if j
rt N
and
*)
If;
= 0 if j EN; i.e., 9 is holomorphic on
V.
It only remains to
check that f is in AO if and only if 9 is in AO(a,H). This is routine and we will do it one way. Assume that f is in AO. Then g(z
+ 1) =
exp ( - 7r
L hizjzj -7r( L hj(z;lj + zili + liTi)) )al iEN
jEN
x exp(7rH(z, 1)
7r
+ 2H(l, l))f(z)
= a, exp (7r H( Z, 1)
+ iH(l, 1) )g( z)
.
So 9 is in AO(a,H).
o
§ 3.6 Examples Let fi' be an invertible sheaf on a complex torus X = V / L. Theorem 3.10 (Riemann-Roch). The Euler characteristic x(fi') of fi' E(-l)idimHi(X,fi') is the intersection number Cl(g~)9 where 9 = dimX.
Proof. Let fi' = fi'(a,H) be some Appel-Humbert data. Then Cl(fi') is the invariant two-form on X corresponding to the skew-symmetric fonn 1m H = E.
§ 3.6 Examples
~~'
27
~,,{ = gfcl(f£)I\(g 1 times)l\cl(f£).
Then by linear algebra ±vuetL E~.f£ll\ ... l\d':&2g Thus H is non-degenerate if and only if
C!
(~). 9
=I o.
Assume that H is degenerate we need to see that X(f£) = O. This case follows from Theorem 3.9a) and b). Because E(-l)iH i (KO(f£),l!'JKO(.2'») = 0 as the cohomology of the structure sheaf is an exterior algebra. If H is nondegenerate then z = 0 and Hi(X,f£) is non-zero when i = n and its dimension is ±JdetLE. Thus we need only check that ± = (_l)n. This is a question in linear alg;ebra which we don't do. (Hint: if H' is a pOf>itive definite form on V when (ImH,)g is positive). 0 A frequently used special case is Corollary 3.11. If D is an ample divisor on an abelian variety X then a) dimT(X,l!'Jx(D)) = the intersection number ~Dg and g. b) Hi(X,l!'Jx(D)) = 0 ifi
> O.
Another special case of Theorem 3.9 is Corollary 3.12. If f£ is in Pico(X) but f£ =1= l!'Jx, then Hi(X,f£) all i.
=0
for
Let ~ be the Poincare sheaf on X x X~ where X~ is dual complex torus. Corollary 3.13. The only non-zero cohomology group of ~ is Hg(X x X~,~) which has dimension one. Proof. We left this as an exercise (Hint: prove that X(~) = ±l and that the tangent space of X(dimg) is an isotropic subspace of H(~)). 0
Next we want to make some applications to families of cohomology groups. Let f : X ~ S be a smooth proper morphism of connected analytic spaces. Assume that the fibers X. = f-l(S) are abelian varieties. Let f£ be an invertible sheaf of X. Theorem 3.14. If for all s in S the sheaf f£lx. is ample on X., then a) f*f£ is a locally free sheaf on S and for each s, (f*f£). ~ T(X., f£IX.) is an isomorphism. b) Rif*f£ = 0 ifi
> O.
28
Chapter 3. The Cohomology of Complex Tori
Proof. Follows from Corollary 3.11 by proper flat base extension. (Actually a) can be proven directly for using the Fourier expansions of sections of !l' along the fibers.) 0
We will need to know one calculation of higher direct images for the Poincare sheaf ~. Theorem 3.15. R g 7r x- o (~) is the one-dimensional sky-scraper sheaf situated at the zero point 0 of X~ and the other higher direct images are zero.
Proof. By proper flat base extension for 7r x~ : X X X~ ~ X~ the Corollary 3.12 implies that support (R i 7r x- o (~)) C {O} for all i. Hence the Leray spectral sequence gives an isomorphism
Thus this result follows from 3.13.
o
Chapter 4. Groups Acting on Complete Linear Systems
§ 4.1 Geometric Background Let !l' be a very ample invertible sheaf on an abelian variety X. Then we have a projective embedding of X in lP n by the complete linear system IDI of sections of !l'.
Lemma 4.1. Translation Tx : X ~ X by a point x of X extends to a projective transformation of IP n if and only if x is contained in the finite group K(!l') = {x E XIT;!l' ~ !l'}.
Proof. We have already seen in Lemma 1.10 that K(!l') is finite. Assume that Tx extends to a projective transformation. Then for any E in IDI the divisor E - x is linearly equivalent to E. As!l' ~ ~Jx(E) and T;!l' ~ 19x (E - x) we must have x in K(!l'). Conversely if a: T;!l' ~ !l'is an isomorphism, then the global effect of a defines an isomorphism IP n ~ IP n which extends the action of Tx on X. 0 As the above extension is clearly unique because X spans lPn, we have a projective representation of the finite abelian group K(!l') on IPn. It will shortly turn out that IP n is a very simple irreducible representation of K(!l') but the purpose of this section is to lay the groundwork for this presentation. Let p: G ~ PGL(n) be a projective representation of a group G. Then we have an exact commutative diagram
1
-----+
C*
-----+
1
-----+
C*
-----+
H
G
!p'
!p
GL(n + 1) ~ PGL(n)
-----+
0
-----+
1
where H is the fiber product of a and p. Thus our projective representation p is determined by the ordinary representation p of the central extension H of G by C*.
30
Chapter 4. Groups Acting on Complete Linear Systems
Now let ft' be an invertible sheaf on X, we want to construct the natural central extension H(ft') of K(ft') by C* together with a representation of H(ft') on T(X, ft') which will give above projective representation of K(ft') on ]pn when ft' is very ample. This group H(ft') is called the theta (or Heisenberg) group of ft'. By definition an clement of H(ft') is a pair (x, a) where x is a point of K(ft') and a is an isomorphism a : T;ft' ~ ft'. The product (y, (3) * (x, a) =
(x
+ y, (30 T;a)
where (30 T;( a) is the composition T;+yft'
= T;(T;ft')
TO( O. By completing the square in the numerator and making a linear change of variable, one is reduced to the case where b = 0 and a = 1. This last case is done by Gauss's well-known polar coordinate trick. 0
Chapter 6. The Algebra of the Theta Functions
§ 6.1
The Addition Formula
Let X = V / L be an abelian variety. We will fix a decomposition L = A ffi B with respect to an ample invertible sheaf !l' = !l'( Ct, H) which is excellent. In this section we will consider a special case of the isogeny Theorem 5.7. Let n be a positive integer. Let Y be the product xn of X with itself n-times. Then we have the sheaf !l'(c) = Q91:5:;:5:n 7rJ!l'0 C; on Y where all C; are positive integers. Clearly !l'(c) is ample and excellent with respect to the product decomposition An ffi Bn of the lattice L n of Y = vn / Ln. Now let d = (d1 , ••• ,dn ) be another such sequence. Let C (resp. D) be the diagonal n x n matrix of integers with entrees Cl, ••. ,C n (resp. d l , ••• ,dn ). Let F be an n x n matrix of integers. Then F define a homomorphism F : Y ~ Y where F(xl> ... ,x n ) = CEI:5:;:5: n FIx;). Lemma 6.1. !l'(c) is isomorphic to F* !l'(d) if and only if t F· D . F = C. In which case F is an isogeny.
Proof. To check that two excellent sheaves are isomorphic it is enough to see that they have the same Hermitian form or what is the same as the same o T(X, J(0 71 ) is generated as a C-algebra by T( X, JI) if J{ is an m-power with m ~ 3. Proof. Without loss of generality we may assume that JI is excellent. In the previous theory let .!l' = J{0 2. For a) we take Xl and X2 such that 2'" ,a g , b1 , ••• ,bg be a canonical basis of L with elementary divisors el, ... ,ego Let e be the diagonal matrix with coefficients el, ... ,ego Then the skew-form E is given by tegral matrix [;
~]
[~e ~].
such that
An element of Sym71(L) is a 2g
~] [~e ~]
t [;
[;
~]
[
X
2g in-
~ e ~ ]. Let
re denote this group. Let b: = -!; b;. Then ai, ... ,ag , bi, . .. ,b~ is a symplectic basis of Wand as such it defines a bijection R: Reas(W) ~ Sg. We know how SymlR(W) acts on B ~ (;g x Sg in tenus of a block decomposition of SymlR(W) with respect to the second basis. If [; element of SymlR (W) is
[~1 e c
e
~]
is in
re
then the corresponding
!Ied] . Thus the corresponding action of B is e
[~ ~] *(Z,T) = ((cT+de)-l ez , (aT+be)(CT+de)-l e). So M(L, W) is isomorphic to re \ Sg with the action given by the second coordinate. given by
§ 7.4 Families of Ample Sheaves on a Variable Abelian Variety In the situation of Section 7.3 we want to consider families of invertible sheaves on the basic family 7r' : B I L ~ Reas(W). First of all we consider the multiplier A/(z) = o:/e+1rHJ(z,/>+fHJ(/,1) where 0:/+/1 = O:r 0:/ , ( _l)E(/,/') as usual. For fixed
64
Chapter 7. Moduli Spaces
J this defines an ample invertible sheaf J{J on X(J). This family does not depend holomorphically on J but only real analytically (recall in coordinates HJ is given by (ImT)-1 where T = R(J)). What we can do is change the horizontal structure on the family J{J so that it is holomorphic. This can be done in many ways depending on the choice of maximal isotropic subgroup A of L. This idea goes back to the proof of Theorem 2.1 in section 2.2. For J in Reas(W), let SA,J be the symmetric complex bilinear form on (w, J) such that SA,J = HJ on A ®71 IR x A ®71 IR. Let ~(A, J) : (W, J) ~ C be the complex linear mapping such that v~(A,J)(w) = E(w,v) for all w in A IR. Assume that a is identically one on A. Then K/,J( v) = a(l)e+ 1ri / (A,J)(/)+21ri/~(A,J)(tI) is factor of automorphy. Let YA,J be the invertible sheaf on X(J) such that multiplication by e{SA,J(tI,tI) gives an isomorphism If'A,J : J{J ~ YA,J.
®71
Lemma 7.8. YA is an invertible sheaf on B I L. In other words the multipliers K/,J(v) are holomorphic for (v,J) in B.
Proof. Let al, . .. ,a g be a basis of A. Then we choose a canonical basis al, . .. ,a g , b1 , ••• ,bg of L. Let ai, ... ,a g , b~, ... ,b~ be the associated symplectic basis of W. g g Then we have the standard isomorphism If' : B ~ cg x Sg and L ~ tl EB tl = EB71aj EB EBbZ( -bj ). Claim. In terms of this isomorphism K/,J is given by
where I
= II EB 12
and
T
= R( J) and z corresponding to v.
This is clear because K/,Av) is one if I is in A by the calculation in the proof of Theorem 2.1. Thus K/ 1 +I"J(v) = K/2,J(V) which can be computed by Lemma 5.2 which determines ~ in terms of T. 0 Next let A be another maximal isotropic subgroup of L such that alA == 1. Then we have the isomorphism K1 == If' A,J 0 1f'"A:J : YA,J ~ YA,J'
Lemma 7.9. K1 is an isomorphismYA ~ Y A of invertible sheaves on BIL.
A' . b ul . l' . b -{(SProo. A,J (tI,tI)-SA ' J(tI.tI» . We nee d to f K A IS given y m tIp lcatIon y e see that the function S A,i v, v) - SA,J( v, v) == QJ( v) is C-analytic in v and J simultaneously. We will do this in coordinates (z, T) with respect to aI, ... ,a g , b1 , ••• ,bg. Let aI, ... ,ii g, b1 , ••• ,b g be another canonical basis of L such that
§ 7.4 Families of Ample Sheaves on a Variable Abelian Variety
A=
fB71aj. Thus we have a symplectic transformation
[~ ~]
65
in Sym71(L)
o
which takes the ordinary basis to the one with the "'. This result will prove Lemma 7.9. Sublemma 7.10. QJCv) is given by -2i tz t"'{ thT complex basis ai, ... ,a g of (W, J).
+ 6e)-l z
in terms of the
Proof. In this basis SA,J(V,V) is given by tz lm (T)-l z by the remarks in the proof of Theorem 5.8. Let r be point in Sg corresponding to J computed with respect to a1,'" ,a g , b1, ... ,bb. Similarly SAjv,v) is given by tZIm(7')-lzwith respect to the complex basis ai, ... ,a g • We know that z = e t( O'T + 6e ) -1 Z and 7' = (aT + fJe)hT + 6e)-l e. We also have the equation
t-I (-)-1-::z= t z Im( T)-1-Z zmT as H J is independent of the coordinates. Now QJ(v) = tZIm(r)-l z - tZ lm (T)-l z which is given in the ordinary bases by the matrix t{ e thT + 6e )-1} Im( (aT + fJe )hT + 6e )-1 e )-1 {e thT + 6e)-1} - (ImT)-l. By (*) we have t{e thT+6e)-1}Im((aT+fJe)hT+6e)-le)-1{e thr+6e)-le}
= (ImT)-l
.
So QJCv) is given by t{e thT + 6e)-1} t{ e thT + 6e)-1 }-l(lm r)-l(e thr + 6)-1 )-1 {e t X
hT+6e)-le}-(lmT)-1
= (1m T )-1 [thr + 6) thT + 6e)-1 - 1] = (ImT)-l [thr + 6) - thT + 6eWhT + 6e)-1 = (1m T )-1 [-2i t( "'{ 1m T) t( "'{T + 6e )-1] = -2i t"'{ t( "'{T + 6e )-1 .
0
Now .-vA has a Hermitian metric so that .. ' ,Bg be a basis of Hl(C, Z) in standard position. Let al>." ,ag, bl>'" ,b g be their dual basis in Hl(C, Z). This is a canonical basis of HI (C, Z). We would like to compute the T-matrix of this Hodge structure. Let WI>'" ,Wg be abelian iategrals normalized such that fBk wi = oj. Then Tt fA Wk. The conditions that T is an element of the Siegel's space Sg are called Riemann's bilinear equations and inequality. Thus P(U) is a principal polarized abelian variety. Hence we have a canonical isomorphism A(U) = PicO(P(U)) and P(U). This abelian variety P(U) is called the Jacobian of C. One must be careful to distinguish between the various equivalent forms of the Jacobian but the autoduality of the Jacobian is an important principle. Next I want to discuss the most direct relationship between the abelian varieties P(U) and A(U) and the curve C. First of all P(U) is the Picard variety Pico(C) of C; i.e. Pico(C) = kerncl(Hl(C, eo) ~ H2(c, Z)) where
e is
2",.-
the boundary in the exact sequence 0 ~ Zc ~ e c c---+ eo ~ O. This follows cohomologically as follows. We have an exact sequence 0 ~ HI (C, Z) ~ Hl(C, ex) ~ Pico(c) ~ 0 by diagram chasing. Now Hl(C, ex) is represented , by the periods of antiholomorphic differentials P and a corresponds to the projection of L on this subspace. For A(P) it is naturally the Albanese variety of C. The complex torus X with a universal analytic mapping S : C ~ X. By Section 9.1 to define S we need to give a C-linear mapping R: Homv(T(C,il),C)* ~ T(C,il) such that for all closed paths 'Y in C, then linear functional x in Hom( T( C, il, C))* defined by f-y R(w) = (x,w) in L.l.; i.e. (X,Wl) + (X,W2) in integral if (WI>W2) is in Hl(C, Z). This is just f-ywi +W2 = b,Wl +W2}. Thus L.l. is the smallest lattice which satisfies the condition. Therefore we have an integral f : C ~ X and it i;; clearly universal.
§ 11.4 Picard and Albanese Varieties for a Kahler Manifold Let Y be a compact Kiihler manifold with Kahler form w. By Hodge Theorem HO(X, ilx) EB HO(X, ilx ) = Hl(X, Z) ®71 C. Thus we have an elementary structure HO(X, ilx ) on Hl(X, Z)/(torsion). Let PicO(X) be P of this Hodge structure and Alb(X) be A of it. Then PicO(X) = {o : Hl(X,e;c) ~ H2(X,Z)} as before as Hl(X,ex) = HO(X,ilx). Similarly with the one dimensional case we have a universal integral f : X ~ Alb(X). One may ask when these tori are abelian. There is a skew-symmetric form E on HI (X, Z) ®71 C given by [0"11\ 0"21\ [w]dim X-l]X. By local calculation we have HO(X,il x ) is isotropic and tE[O",a-] is positive definite of HO(X,il x ). Thus we have all the conditions but that E is integral on HI(X, Z)/(torsion).
100
Chapter 11. Abelian Varieties Occurring in Nature
Now Y is projective variety and w represents first Chern class of an ample line bundle. Then w and hence IE is integral. Therefore we have proven
Proposition 11.9. The Picard and Albanese of a smooth projective vanety are a dual pair of abelian varieties.
Informal Discussions of Immediate Sources
The statement of the Appel-Humbert Theorem 1.5 is due to Mumford [3J. Also I follow his idea for the proof of the existence Theorem 2.2. Some of the material in Chap. 1 is to introduce the reader to the abstract statemcnts used since A. Wei! in the algebraic casco The Scct. 3.4 is modeled on H. Langes lecturcs notes wherc he uses idea::. from Umemura [8J. Thc idea of Chap. 4 comes from Mumford's famous paper [4J. Lemma 4.6 is founded in [5J. The Theorem 5.9 and its application in Chap. 8 to the functional equation of the theta function can be found in Igusa [lJ. The algebraic material is Chap. 6 was initiated by Mumford [4J, added to by Koizumi and Sekiguchi and completed in [2J. Section 8.3 is close to Mumford's discussion in [6J. Proposition 9.7 is due to Serrc. Proposition 9.10 is due to Z. Ran. Scction 10.1 to 10.3 are an adaptation of H. Lange's notes again. He attributes Theorem 10.1 to Obuki. The Theorem 10.6 is due to Sasaki by a different method. Lange and Narasimhan have informed me that they have another proof. The point of vicw of Chap. 11 is from that of Deligne's idea of a Hodge structure.
References
[1] J.-1. Igusa: Theta Functions. Springer, New York 1972 [2] G. Kempf: Projective Coordinate Rings of Abelian Varieties. Algebraic Analysis, Geometry and Number Theory. Edited by J.1. Igusa, Johns Hopkins Press 1989, pp.225-236 [3] D. Mumford: Abelian Varieties. Oxford University Press, Oxford 1970 [4] D. Mumford: On the Equations Defining Abelian Varieties. Invent. math. 1 (1966) 287-354 [5] D. Mumford: Varieties Defined by Quadratic Equations. In: Questions on Algebraic Varieties. Centro. Intern. Mate., Estrivo, Roma, 1970, pp.31-100 [6] D. Mumford: Tata l.€ctures on Theta 1. Prog. in Math., Vol. 28. Birkhauser, Boston 1983 [7] C.L. Siegel: Symplectic Geometry. Academic Press, New York 1964 [8] H. Umemura: Nagoya Math. J. 52 (1983) 97-128
Subject Index
Abelian variety 13 Addition Theorem 46 Albanese variety 99 Appel-Humbert data, 4 Theorem 4 Complex torus 1 ExceIlent sheaf 38 Factors of automorphy 4 Hodge structures 95 Isogeny 2 Jacobian 98
ImDI
where m :2: 3 16 m=2 88 Modular form 69 Mumford's Embedding Theorem 78
PicO(V/ L) 5 Picard variety 99 Poincare sheaf B', 6 cohomology of 27 r.pz
7
Reas(W) 55 Riemann-Roch Theorem 26 Sheaf fi'( a, H), 4 sections of 9 matrix on 9 cohomology of 25 Siegel space Sy 58 Theorem of the cube 8 Theorem of square 14 Theory of theta functions 35 length of 35 analytic theory 39 Theta group H(fi') 29
