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4.4. JACOBIANS 155

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
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

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.

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
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).

THEOREM 4.4.3 (i) Any abelian variety is a complex projective
(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).
(Since we know that a K¨hler form ω on CPN has this property.) Now
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
THEOREM 4.4.4 Let (X, J) be a compact K¨hler manifold. The
following are equivalent
(i) X is projective
(ii) There exists a K¨hler structure ω on (X, J) such that the cohomol-
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,
this form ω will restrict to an integral K¨hler form on X. This proves
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

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
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.

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.
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´:
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.

Proof. (Sketch.) So we have
Cg /“
∪ ,
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


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