1. THE CLASSICAL THEORY: PART I 7
normalize to have π1 = 1, so that setting τ = π2 the normalized period matrix is
[
τ
1
] where Im τ 0.
τ
1
Differential forms on an n-dimensional complex manifold Y with local holo-
morphic coordinates z1,...,zn are direct sums of those of type (p, q)
f dzi1 · · · dzip
p
d¯j1 z · · · d¯jq z
q
.
Thus the C∞ forms of degree r on Y are

⎨Ar(Y
⎩Aq,p(Y
) =
p+q=r
Ap,q(Y
)
)= Ap,q(Y ).
Setting
H1,0(X)
= span{dz}
H0,1(X)
= span{d¯} z
we have
H1(X, C)= H1,0(X) H0,1(X)
H0,1(X)
= H1,0(X).
This says that the above decomposition of the 1-forms on X induces a similar
decomposition in cohomology. This is true in general for a compact ahler manifold
(Hodge’s theorem) and is the basic starting point for Hodge theory. A recent source
is [Cat1].
From dz dz = 0 and
(
i
2
)
dz z 0, by using that cup-product is given in
de Rham cohomology by wedge product and integration over X we have
Q
(
H1,0(X),H1,0(X)
)
= 0
iQ
(
H1,0(X), H1,0(X)
)
0.
Using the above bases the matrix for Q is
Q =
0 −1
1 0
and these relations are
Q(Π, Π) = tΠQΠ = 0
iQ(Π, Π) =
itΠQΠ
0.
For Π = [
τ
1
] the second is just Im τ 0.
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