Moreover, Hg /Sp(2g, Z) has the structure of locally closed algebraic

subvariety of CPN for some N . It is de¬ned by algebraic equations

with integer coe¬cients. Besides, the fundamental domain F is path-

√

connected. This follows from the observation that if Zi = Xi + ’1Yi ,

√

i = 1, 2 belong to F then Zi = Xi + »i ’1Yi do too for »i > 1 and

suitably choosing »1 , »2 we may connect Z1 and Z2 by a line segment.

Let us give another description of Hg . Namely that Hg is the space

of complex structures J on R2g compatible with the standard symplectic

structure ω such that B(u, v) = ω(Ju, v) is a positive de¬nite symmetric

bilinear form. √ is often called the set of K¨hler polarizations of R2n .

It a

For Z = X + ’1Y we have

Y ’1 X ’Y ’1

Z’J = .

¯

’ZY ’1 Z Y ’1 X

Let us brie¬‚y explain how this matrix comes about. J is the endomor-

phism of square ’Id for which the eigenspace of i (resp. ’i) is the

¯

Z Z

column space of the matrix , resp. . Therefore we have

Id Id

iId 0

M ’1 .

J =M

0 ’iId

Let us denote Ag = Hg /Sp(2g, Z), which is the same as the double

coset space U (g) \ Sp(2g, R)/Sp(2g, Z). There is a strong result due

to Satake that Ag is a locally closed algebraic subvariety of CPN for

some N , which means that it is de¬ned by P1 = · · · = Pk = 0, Q = 0,

where Pi and Q are some polynomials in the homogeneous coordinates

on CPN .

4.4 Jacobians

As we already know, to each Riemann surface Σ of genus g we can

associate a point in Ag . But also, to each such Σ we can associate

a complex torus J(Σ) of dimension g called the Jacobian variety of

Cg /Λ, where

Σ or simply Jacobian. The complex manifold J(X)

Λ is a cocompact lattice in Cg is constructed as follows. As Cg we

take the space „¦1 (Σ) = Hom(„¦1 (Σ), C) - the dual to the space of the

156 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

holomorphic one-forms on Σ. As Λ one takes H1 (Σ, Z), viewing an class

[γ] ∈ H1 (Σ, Z) of a cycle γ as a linear functional φγ on „¦1 (Σ):

φγ (ω) = ω.

γ

One uses the following arguments to see that H1 (Σ, Z) is a cocompact

lattice in „¦1 (Σ) . Consider the commutative diagram

H1 (Σ, Z) ’ H1 (Σ, R)

“ ±

„¦1 (Σ)

The group H1 (Σ, Z) is a cocompact lattice in H1 (Σ, R) and the map

± is an isomorphism: this is because the two spaces have the same

dimension and if [γ] ∈ H1 (Σ, R) is perpendicular to every vector from

„¦1 (Σ) then it is also perpendicular to any vector from „¦1 (Σ), thus to

the whole H 1 (Σ, C), which contradicts to the duality between H1 and

H 1.

Now we de¬ne J(Σ) = „¦1 (Σ) /H1 (Σ, Z). The tangent space to the

Jacobian at the identity O can be identi¬ed with

„¦1 (Σ)

TO J(Σ) H1 (Σ, R) H1 (Σ, Z) —Z R.

The skew-symmetric non-degenerate intersection pairing E on H1 (Σ, R)

de¬nes a translation invariant K¨hler form on J(Σ). It also has the

a

following three important properties.

1). E takes integer values on the lattice H1 (Σ, Z) ‚ H1 (Σ, R)

2). The lattice H1 (Σ, Z) is self-dual with respect to E

3). E = Im(H) for some positive de¬nite hermitian form H

The ¬rst property is equivalent to the fact that E de¬nes an integral

cohomology class in H 2 (J(Σ), R).

Let us explain what is the meaning of the second property. If A ‚

H1 (Σ, R) is a cocompact lattice, and if E is a bilinear form, then the dual

lattice A consists of such vectors v ∈ H1 (Σ, R) that satisfy E(v, a) ∈ Z)

for any a ∈ A. Clearly, (A ) = A and (»A) = »’1 A . Besides, E takes

integer values on A if and only if A ‚ A . Let det(A) stand for the

volume of the quotient H1 (Σ, R)/A with respect to the volume form

given by E. If E restricted to A takes only integer values, then the

4.4. JACOBIANS 157

group A /A has cardinality equal to (det A)’2 . Therefore A is self-dual

if and only if | det A| = 1 which in turn happens if only if the matrix of

E with respect to a basis of A has determinant equal to one.

Let us also show that the third property holds true for the form

E on H1 (Σ, R), since it is not entirely obvious. If (V, J) is a complex

vector space then we decompose

¯

V —R C = Vh • Vh

√ √

according to the eigenvalues ’1 and ’ ’1 of J respectively. Also, if

E is a real-valued skew-symmetric bilinear form on V then it is possible

to extend E by a complex-valued bilinear form on V — C.

LEMMA 4.4.1 The following are equivalent

(i) E = Im(H) for some positive de¬nite hermitian form H

(ii) Vh is a lagrangian subspace with respect to E and we have

√

’1E(v, v ) > 0,

¯

for any non-zero v ∈ Vh .

Proof. We have

√ √

H(ξ, ·) = E(Jξ, ·) + ’1E(ξ, ·) = B + ’1E,

where B is symmetric. We saw before that any v ∈ Vh can be repre-

√ √

sented as v = ξ ’ ’1Jξ for some ξ ∈ V , then also v = ξ + ’1Jξ.

¯

Then we have

√ √

E(v, v ) = 2 ’1E(ξ, Jξ) = ’2 ’1B(ξ, ξ)

¯

and √

’1E(v, v ) = 2B(ξ, ξ) = 2H(ξ, ξ).

¯

We also notice that the map V ’ Vh given by v ’ ξ is an isomorphism.

DEFINITION 4.4.2 An abelian variety is a complex torus V /“ (“ is

a cocompact lattice in a complex vector space V ) for which there exists

a skew-symmetric real-valued bilinear form E satisfying 1). and 3).

above

158 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

THEOREM 4.4.3 (i) Any abelian variety is a complex projective

manio¬‚d.

(ii) Conversely, if a complex torus is a projective manifold, it is an

abelian variety.

Proof. We do not prove part (i) as the classical proof uses the theory

of theta-functions, a topic which we do not develop here. We refer the

reader to the book of A. Weil and Igusa.

(ii) By assumption, there is an embedding Cg /Λ ’ CPN , hence

Cg /Λ is a K¨hler manifold and it has a K¨hler structure for which the

a a

K¨hler form β de¬nes an integer cohomology class: β ∈ H 2 (Cg /Λ, Z).

a

(Since we know that a K¨hler form ω on CPN has this property.) Now

a

we average β under the translations of Cg /Λ:

—

E= Tx (β)dµ(x),

Cg /Λ

where µ is the Haar measure of total volume 1. It is now quite simple

to see that E is translation invariant and satis¬es 1). and 3).

Sometimes it is very useful to have the following criterion due to

Kodaira

THEOREM 4.4.4 Let (X, J) be a compact K¨hler manifold. The

a

following are equivalent

(i) X is projective

(ii) There exists a K¨hler structure ω on (X, J) such that the cohomol-

a

ogy class ω is integral.

We prove (i) implies (ii). First, we observe that a class [ω] is integral if

and only if every period γ ω, where [γ] ∈ H1 (X, Z) is an integer. An-

other basic fact is that if X is projective, then there exists an embedding

X ’ CPN for some N . The projective space CPN , as we saw, admits

a K¨hler structure ω such that [ω] = 1 ∈ H 2 (CPN , Z) Z. Therefore,

a

this form ω will restrict to an integral K¨hler form on X. This proves

a

that (i) follows (ii). To get the converse statement, Kodaira starts with

a K¨hler manifold (X, J, ω) and produces a holomorphic line bundle

a √

L on X with a connection, such that its curvature is 2π ’1ω. Then

vanishing theorems follow that for a large M the line bundle L—M has

4.4. JACOBIANS 159

lots of holomorphic sections. Let us take a basis (s1 , ..., sN +1 ) of holo-

morphic sections of L—M and let φ(x) = [s1 (x) : · · · : sN +1 (x)] be the

desired map φ : X ’ CPN . The proof of (ii) ’ (i) is given in the book

of Gri¬ths and Harris [31].

DEFINITION 4.4.5 The symplectic form E is called a polarization

of the complex torus. If, in addition, the lattice is self-dual, E is called

a principal polarization. In the latter case we call the complex torus

satisfying 1), 2), and 3) above a principally polarized abelian variety

(PPAV).

For example, if Σ is a Riemann surface, then the jacobian variety J(Σ)

is PPAV.

DEFINITION 4.4.6 The moduli space X of PPAV is a variety sat-

isfying the following two conditions.

- A point of X is an isomorphism class of PPAV.

- Given any holomorphic family f : Y ’ B of PPAV, the map

B ’ X: b ’ [f ’1 (b)] is holomorphic.

This notion of moduli space is what is called a coarse moduli space.

In fact, the conditions we put are even weaker than those for a coarse

moduli space.

THEOREM 4.4.7 The space

Ag = Hg /Sp(2g, Z)

is the moduli space of PPAV.

We provide the reader with a brief geometric summary of related

results. Let Mg be the moduli space of compact Riemann surfaces of

genus g > 0. It has dimension equal to 3g ’3+δ1g . There is the Torelly

map

T : Mg ’ Ag , T (Σ) = J(Σ),

which is injective and holomorphic. The basic problem which appears

here is to generalize the Torelli map to higher-dimensional complex

manifolds and the basic tool to deal with it is Hodge theory.

160 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

We notice that if G is a connected compact complex Lie group, then

G = Cg /“. To see this, one needs ¬rst to establish commutativity of G.

Let g be the Lie algebra of G. The commutativity of G follows from

Liouville theorem applied to the adjoint representation G ’ Aut(g).

Now, when G is commutative, the exponential map

exp : g ’ G

is a group homomorphism and we let “ be its kernel. The group “ is dis-

crete, because exp is a di¬eomorphism in a neighbourhood of 0 ∈ g and

the inverse function theorem follows that there exists a neighbourhood

U of 0 such that “ © U = {0}.

Let us give some examples of abelian varieties. The very basic one

is the example of an elliptic curve X = C/(Z • „ Z), where Im(„ ) > 0.

Taking its products, we get more examples A = X — · · · — X. If E is

a polarization of X then E • · · · • E would be a polarization of A.

Besides,

H1 (A, R) = H1 (X, R) • · · · • H1 (X, R).

Next we introduce the notion of isogeny for abelian varieties, and

more general, for complex tori. Let us have two cocompact lattices

“ ‚ “ ‚ Cg so that “/“ is a ¬nite abelian group. Then there is

naturally a ¬nite Galois covering

φ : Cg /“ ’ Cg /“

with the ¬bers “/“ and the group “/“ acts on Cg /“ and this action

induces the identity on Cg /“. Such a map φ is called an isogeny. In the

given situation, if one of two tori admits a polarization, then so does

the other.

LEMMA 4.4.8 Isogeny is an equivalence relation.

Proof. If “ ‚ “ then there exist an integer n such that n“ ‚ “ . Now

the two tori Cg /n“ and Cg /“ are isomorphic, and we leave the rest as

a simple exercise for the reader.

A much deeper result is due to Poincar´:

e

THEOREM 4.4.9 Let A be an abelian variety and let A be an abelian

subvariety, then there exists another abelian subvariety B such that A

is isogeneous to the product A — B.

4.5. ALGEBRAIC CYCLES 161

Proof. (Sketch.) So we have

Cg /“

A=

∪ ,

A = V /V © “

where V is a subspace of Cg . The subgroup Λ = V © “ of “ has the √

property that its real span is stable under the multiplication by ’1.

Tensoring with Q we get Λ — Q ‚ “ — Q. It is clear that the polarization

E restricts to Λ—Q to give a non-degenerate form, the imaginary part of

a positive-de¬nite hermitian form. Thus we can formally get orthogonal