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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 deп¬Ѓned by algebraic equations
with integer coeп¬ѓcients. Besides, the fundamental domain F is path-
в€љ
connected. This follows from the observation that if Zi = Xi + в€’1Yi ,
в€љ
i = 1, 2 belong to F then Zi = Xi + О»i в€’1Yi do too for О»i > 1 and
suitably choosing О»1 , О»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 П‰ such that B(u, v) = П‰(Ju, v) is a positive deп¬Ѓnite symmetric
bilinear form. в€љ is often called the set of KВЁhler polarizations of R2n .
It a
For Z = X + в€’1Y we have

Y в€’1 X в€’Y в€’1
Zв†’J = .
ВЇ
в€’ZY в€’1 Z Y в€’1 X

Let us brieп¬‚y explain how this matrix comes about. J is the endomor-
phism of square в€’Id for which the eigenspace of i (resp. в€’i) is the
ВЇ
Z Z
column space of the matrix , resp. . Therefore we have
Id Id
iId 0
M в€’1 .
J =M
0 в€’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 deп¬Ѓned by P1 = В· В· В· = 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 ОЈ of genus g we can
associate a point in Ag . But also, to each such ОЈ we can associate
a complex torus J(ОЈ) of dimension g called the Jacobian variety of
Cg /О›, where
ОЈ or simply Jacobian. The complex manifold J(X)
О› is a cocompact lattice in Cg is constructed as follows. As Cg we
take the space в„¦1 (ОЈ) = Hom(в„¦1 (ОЈ), C) - the dual to the space of the
156 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

holomorphic one-forms on ОЈ. As О› one takes H1 (ОЈ, Z), viewing an class
[Оі] в€€ H1 (ОЈ, Z) of a cycle Оі as a linear functional П†Оі on в„¦1 (ОЈ):

П†Оі (П‰) = П‰.
Оі

One uses the following arguments to see that H1 (ОЈ, Z) is a cocompact
lattice in в„¦1 (ОЈ) . Consider the commutative diagram

H1 (ОЈ, Z) в†’ H1 (ОЈ, R)
в†“ О±
в„¦1 (ОЈ)

The group H1 (ОЈ, Z) is a cocompact lattice in H1 (ОЈ, R) and the map
О± is an isomorphism: this is because the two spaces have the same
dimension and if [Оі] в€€ H1 (ОЈ, R) is perpendicular to every vector from
в„¦1 (ОЈ) then it is also perpendicular to any vector from в„¦1 (ОЈ), thus to
the whole H 1 (ОЈ, C), which contradicts to the duality between H1 and
H 1.
Now we deп¬Ѓne J(ОЈ) = в„¦1 (ОЈ) /H1 (ОЈ, Z). The tangent space to the
Jacobian at the identity O can be identiп¬Ѓed with

в„¦1 (ОЈ)
TO J(ОЈ) H1 (ОЈ, R) H1 (ОЈ, Z) вЉ—Z R.

The skew-symmetric non-degenerate intersection pairing E on H1 (ОЈ, R)
deп¬Ѓnes a translation invariant KВЁhler form on J(ОЈ). It also has the
a
following three important properties.
1). E takes integer values on the lattice H1 (ОЈ, Z) вЉ‚ H1 (ОЈ, R)
2). The lattice H1 (ОЈ, Z) is self-dual with respect to E
3). E = Im(H) for some positive deп¬Ѓnite hermitian form H
The п¬Ѓrst property is equivalent to the fact that E deп¬Ѓnes an integral
cohomology class in H 2 (J(ОЈ), R).
Let us explain what is the meaning of the second property. If A вЉ‚
H1 (ОЈ, R) is a cocompact lattice, and if E is a bilinear form, then the dual
lattice A consists of such vectors v в€€ H1 (ОЈ, R) that satisfy E(v, a) в€€ Z)
for any a в€€ A. Clearly, (A ) = A and (О»A) = О»в€’1 A . Besides, E takes
integer values on A if and only if A вЉ‚ A . Let det(A) stand for the
volume of the quotient H1 (ОЈ, 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)в€’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 (ОЈ, R), since it is not entirely obvious. If (V, J) is a complex
vector space then we decompose
ВЇ
V вЉ—R C = Vh вЉ• Vh
в€љ в€љ
according to the eigenvalues в€’1 and в€’ в€’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 вЉ— C.
LEMMA 4.4.1 The following are equivalent
(i) E = Im(H) for some positive deп¬Ѓnite hermitian form H
(ii) Vh is a lagrangian subspace with respect to E and we have
в€љ
в€’1E(v, v ) > 0,
ВЇ
for any non-zero v в€€ Vh .
Proof. We have
в€љ в€љ
H(Оѕ, О·) = E(JОѕ, О·) + в€’1E(Оѕ, О·) = B + в€’1E,
where B is symmetric. We saw before that any v в€€ Vh can be repre-
в€љ в€љ
sented as v = Оѕ в€’ в€’1JОѕ for some Оѕ в€€ V , then also v = Оѕ + в€’1JОѕ.
ВЇ
Then we have
в€љ в€љ
E(v, v ) = 2 в€’1E(Оѕ, JОѕ) = в€’2 в€’1B(Оѕ, Оѕ)
ВЇ
and в€љ
в€’1E(v, v ) = 2B(Оѕ, Оѕ) = 2H(Оѕ, Оѕ).
ВЇ
We also notice that the map V в†’ Vh given by v в†’ Оѕ is an isomorphism.

DEFINITION 4.4.2 An abelian variety is a complex torus V /О“ (О“ 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
manioп¬‚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 /О› в†’ CPN , hence
Cg /О› is a KВЁhler manifold and it has a KВЁhler structure for which the
a a
KВЁhler form ОІ deп¬Ѓnes an integer cohomology class: ОІ в€€ H 2 (Cg /О›, Z).
a
(Since we know that a KВЁhler form П‰ on CPN has this property.) Now
a
we average ОІ under the translations of Cg /О›:
в€—
E= Tx (ОІ)dВµ(x),
Cg /О›

where Вµ is the Haar measure of total volume 1. It is now quite simple
to see that E is translation invariant and satisп¬Ѓes 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 KВЁhler manifold. The
a
following are equivalent
(i) X is projective
(ii) There exists a KВЁhler structure П‰ on (X, J) such that the cohomol-
a
ogy class П‰ is integral.
We prove (i) implies (ii). First, we observe that a class [П‰] is integral if
and only if every period Оі П‰, where [Оі] в€€ H1 (X, Z) is an integer. An-
other basic fact is that if X is projective, then there exists an embedding
X в†’ CPN for some N . The projective space CPN , as we saw, admits
a KВЁhler structure П‰ such that [П‰] = 1 в€€ H 2 (CPN , Z) Z. Therefore,
a
this form П‰ will restrict to an integral KВЁhler form on X. This proves
a
that (i) follows (ii). To get the converse statement, Kodaira starts with
a KВЁhler manifold (X, J, П‰) and produces a holomorphic line bundle
a в€љ
L on X with a connection, such that its curvature is 2ПЂ в€’1П‰. Then
vanishing theorems follow that for a large M the line bundle LвЉ—M has
4.4. JACOBIANS 159

lots of holomorphic sections. Let us take a basis (s1 , ..., sN +1 ) of holo-
morphic sections of LвЉ—M and let П†(x) = [s1 (x) : В· В· В· : sN +1 (x)] be the
desired map П† : X в†’ CPN . The proof of (ii) в‡’ (i) is given in the book
of Griп¬ѓths and Harris .

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 ОЈ is a Riemann surface, then the jacobian variety J(ОЈ)
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 в†’ B of PPAV, the map
B в†’ X: b в†’ [f в€’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 в€’3+Оґ1g . There is the Torelly
map
T : Mg в†’ Ag , T (ОЈ) = J(ОЈ),
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 /О“. To see this, one needs п¬Ѓrst 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 в†’ Aut(g).
Now, when G is commutative, the exponential map
exp : g в†’ G
is a group homomorphism and we let О“ be its kernel. The group О“ is dis-
crete, because exp is a diп¬Ђeomorphism in a neighbourhood of 0 в€€ g and
the inverse function theorem follows that there exists a neighbourhood
U of 0 such that О“ в€© U = {0}.
Let us give some examples of abelian varieties. The very basic one
is the example of an elliptic curve X = C/(Z вЉ• П„ Z), where Im(П„ ) > 0.
Taking its products, we get more examples A = X Г— В· В· В· Г— X. If E is
a polarization of X then E вЉ• В· В· В· вЉ• E would be a polarization of A.
Besides,
H1 (A, R) = H1 (X, R) вЉ• В· В· В· вЉ• 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
О“ вЉ‚ О“ вЉ‚ Cg so that О“/О“ is a п¬Ѓnite abelian group. Then there is
naturally a п¬Ѓnite Galois covering
П† : Cg /О“ в†’ Cg /О“
with the п¬Ѓbers О“/О“ and the group О“/О“ acts on Cg /О“ and this action
induces the identity on Cg /О“. Such a map П† 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 О“ вЉ‚ О“ then there exist an integer n such that nО“ вЉ‚ О“ . Now
the two tori Cg /nО“ and Cg /О“ are isomorphic, and we leave the rest as
a simple exercise for the reader.
A much deeper result is due to PoincarВґ:
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 Г— B.
4.5. ALGEBRAIC CYCLES 161

Proof. (Sketch.) So we have
Cg /О“
A=
в€Є ,
A = V /V в€© О“
where V is a subspace of Cg . The subgroup О› = V в€© О“ of О“ has the в€љ
property that its real span is stable under the multiplication by в€’1.
Tensoring with Q we get О› вЉ— Q вЉ‚ О“ вЉ— Q. It is clear that the polarization
E restricts to О›вЉ—Q to give a non-degenerate form, the imaginary part of
a positive-deп¬Ѓnite hermitian form. Thus we can formally get orthogonal
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