<<

. 24
( 44 .)



>>

As the next step, we blow up X at those 16 points to get a non-
˜
singular variety X. The operation of blowing up one point can be
analyzed in our local model: it transforms the cone into a cylinder, and
the vertex of the cone corresponds to a projective conic in the cylinder.
˜ ˜
This conic is called an exceptional line in X. In fact X is K¨hler . We
a
will show that
˜ ˜
H 0 (X, C) = C = H 4 (X, C)
˜ ˜
H 1 (X, C) = H 3 (X, C) = 0
˜
H 2 (X, C) = H 2 (X, C) • C16 ,
where C16 is spanned by the cohomology classes of those exceptional
lines.
LEMMA 4.3.2 Let M be a manifold and let G be a ¬nite group of
automorphisms of M. Then

H — (M/G, C) = H — (M, C)G .
144 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

Thus the cohomology H — (X, C) is computed by the complex (A— (M )G , d),
where A— (M )G is the space of complex-valued G-invariant di¬erential
forms on M. Thus H — (X, C) = H — (A, C)ι . On H 1 (A, C) = Hom(Λ, C)
the map ι also acts as ’Id. Hence on H j (A, C) = §j H 1 (A, C) it acts
as (’1)j . From here we are able to conclude that

H j (X, C) = H j (A, C), j is even;

H j (X, C) = 0, j is odd.
It means that we have zero cohomology for X in dimensions 1 and
3, besides H 2 (X, C) = H 2 (A, C) has complex dimension 6. Using the
Hodge decomposition

H 2 (A, C) = H 2,0 (A) • H 1,1 (A) • H 0,2 (A)

˜ ˜
we see immediately that h2,0 (X) = h2,0 (A) = 1, h1,1 (X) = h1,1 (A) +
˜
16 = 20, and h0,2 (X) = 1. One can ask what is the meaning of
˜
h2,0 (X) = 1 ? It means that there exists unique holomorphic 2-form β
˜
on X which comes from the holomorphic 2-form dz1 §dz2 on A. Further-
more, local computations at ¬xed points show that β never vanishes.
˜
(In the late 70s Yau showed that X has a hyper-K¨hler structure.)
a
The Kummer surface that we considered is an example of a K3 surface,
which all enjoy the property of having locally non-vanishing holomor-
phic 2-form.

Now we go one dimension down and consider a compact Riemann
surface Σ of genus g. As we did before, we pick a nice basis of inte-
ger homology: a1 , ..., ag , b1 , ..., bg , such that the intersection pairing on
H1 (Σ, Z) gives

(ai , aj ) = (bi , bj ) = 0, (ai , bj ) = δij .

The linear span < a1 , ..., ag > is a lagrangian subspace of H1 (Σ, Z). We
saw before the symplectic pairing on cohomology H 1 (Σ, C) given by

Q([±], [β]) = ±§β.
Σ
4.3. EXAMPLES AND SIEGEL SPACE 145

In terms of periods it is given by
g
Q([±], [β]) = ( ±)( β) ’ ( ±)( β).
ai bi bi ai
i=1

We want to use the complex structure on Σ in a signi¬cant way. We
know that

H 1 (Σ, C) = H 1,0 (Σ) • H 0,1 (Σ) = „¦1 (Σ) • „¦1 (Σ).

We also know that „¦1 (Σ) is a lagrangian subspace of H 1 (Σ, C).
LEMMA 4.3.3 There exists a unique (normalized) basis (ω1 , ..., ωg )
of „¦1 (Σ) such that aj ωi = δij .

Proof. Let us take any basis (·1 , ..., ·g ) of „¦1 (Σ). We will show that
the matrix ( ai ·j ) is non-singular. After that one can apply a linear
transformation to make it the identity matrix. This amounts to showing
that if ω ∈ „¦1 (Σ) is such that ai ω = 0 for any i then ω = 0. We
remember that the space V of γ ∈ H 1 (Σ, C) such that ai γ = 0 for
any i is a lagrangian subspace. This subspace is de¬ned over the real
numbers, i.e. V is stable under complex conjugation. Now ω ∈ V
implies that ω ∈ V , and since V is lagrangian we have
¯

ω§¯ = 0,
ω
Σ

which contradicts to our previous knowledge.

Let us consider in detail the genus 1 case. We identify Σ = C/Λ,
where Λ is a lattice spanned by complex numbers w1 and w2 satisfying
Im(w2 /w1 ) > 0. As a basis in H1 (Σ, Z) we can take loops a and b
obtained from the line segments connecting the origin with the points
w1 and w2 respectively. The periods of the holomorphic one-form dz
are
dz = w1 , dz = w2 .
a b

The invariant of the Riemann surface Σ is the ratio „ = w2 /w1 , which
satis¬es Im(„ ) > 0. One can also rescale the lattice Λ in such a way
that w1 = 1.
146 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

Let us denote
Z=( ωi )
bj

the g — g period matrix.

THEOREM 4.3.4 (Riemann-Siegel) (i) Z is a symmetric matrix
(ii) Im(Z) is positive-de¬nite.

Proof. We note that

Q([ωi ], [ωj ]) = 0 = ( ωi )( ωj ) ’ ( ωi )( ωj ) =
ak bk bk ak
k


= ωj ’ ωi = zij ’ zji .
bi bj

For the second part we recall that the hermitian form H(u, v) = ’1Q(u, v )
¯
is positive de¬nite. Let us compute the matrix of H in the basis (ωi ).
√ √
Hij = H(ωi , ωj ) = ’1( ωj ’
¯ ωi ) = ’1(¯ij ’ zij ) = 2Im(zij ).
z
bi bj

Here we used ωi = 1.
ai

DEFINITION 4.3.5 The Siegel upper-half space Hg is the space of
complex g — g matrices satisfying conditions (i) and (ii) of the above
Theorem.

We will show that the space Hg is a homogeneous space of symplectic
group.
To see the geometry of Hg let us consider the genus 2 case. Here
H2 = R3 — U , where R3 corresponds to real symmetric 2 — 2 matrices
and U corresponds to real symmetric positive de¬nite 2 — 2 matrices.
In general, Hg is a subset of complex symmetric g — g matrices and
it looks like an open convex cone. Another geometric interpretation of
Hg is given by

PROPOSITION 4.3.6 The space Hg identi¬es with the manifold S
of complex lagrangian subspaces Λ ‚ C2g such that

’1Q(v, v ) > 0
¯
4.3. EXAMPLES AND SIEGEL SPACE 147

for any v ∈ Λ, v = 0 . Here C2g has the standard symplectic form given
by
0 ’Id
.
Id 0
Proof. The space S lies inside the grassmanian Gras(g, 2g) of complex
g-dimensional subspaces inside 2g-dimensional complex space. The
map Hg ’ S we are looking for is given by

Z
Z ’ column space of ‚ S.
Id

To get the inverse map S√ Hg we take Λ ∈ C2g - a lagrangian subspace

with the property that ’1ω(v, v ) > 0 for any v ∈ Λ. We can view
¯
A
Λ as a column space of some 2g — g matrix , where A and B
B
are invertible as we saw before. This matrix can be replaced by the
equivalent matrix

AB ’1
A Z
B ’1 = = .
B Id Id

The matrix Z automatically satis¬es to the above conditions. For ex-
ample, when Λ is a lagrangian subspace we get

0 ’Id Z
= (t Z ’ Z) = 0,
( tZ Id )
Id 0 Id

implying that Z is symmetric.

There is an action of the complex symplectic group Sp(2g, C) on the
set of lagrangian subspaces, but this action does not preserve positivity
condition. The action of the real symplectic group Sp(2g, R) though
has all the necessary properties. Let us exhibit how γ ∈ Sp(2g, R) acts
AB
on an element of S. Let γ = be represented as four square
CD
blocks. Then we have
(AZ + B)(CZ + D)’1
AB Z AZ + B
= ∼ .
CD Id CZ + D Id
148 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

PROPOSITION 4.3.7 There is a holomorphic action of Sp(2g, R)
on Hg such that

A B
· Z = (AZ + B)(CZ + D)’1 .
C D

The action is transitive and the stabilizer of ’1 · Id is the unitary
group U (g) ‚ Sp(2g, R).

We notice that all stabilizers in Sp(2g, R) of all points of Hg have the
same dimension. We identify Hg S and consider an element V ∈ S,
¯
where V is a lagrangian subspace in C2g . We have C2g = V • V . If
we think of C2g as the complexi¬cation of R2g then R2g has a complex
√ √
¯
structure J which acts as ’1 on V and as ’ ’1 on V . The stabilizer
HV of V is then the set of real linear symplectic transormations γ
which commute with J. It happens if and only if γ is a complex-
linear transformation. Thus Hv U (g). When we compute the real
dimension of the orbit of V it is equal to

dimR Sp(2g, R) ’ dimR U (g) = dimR Hg .

This shows that there is only one orbit, which proves the transitivity.


COROLLARY 4.3.8 The space Hg identi¬es with the coset space

Sp(2g, R)/U (g).

We also notice that

U (g) = Sp(2g, R) © SO(2g, R)

is a maximal compact subgroup in Sp(2g, R). Therefore Hg is a so-
called symmetric space, which by de¬nition is a riemannian manifold
such that for any point Z ∈ Hg there is a symmetry σ : Hg ’ Hg
which ¬xes Z and has Z as isolated ¬xed point (hence the di¬erential

of σ at Z is ’Id). For example the symmetry for Z = ’1Id is
Z ’ ’Z ’1 . The Lie group Sp(2g, R) is the group of isometries of Hg ,
4.3. EXAMPLES AND SIEGEL SPACE 149

and its action is transitive. In addition, Hg is a K¨hler manifold under
a
a K¨hler potential
a

ρ(Z) = log(det(Im(Z))).

For g = 1 and the coordinate z = x + ’1y on the Lobachevsky plane
we have recover ρ(z) = log(y).

<<

. 24
( 44 .)



>>