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 K¨ 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 ∧ d¯ 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.