6 1. THE CLASSICAL THEORY: PART I

is a lattice. The pictures are

δ2

δ1

π2

π1

Here δ1 ↔ π1 and δ2 ↔ π2 give a basis for H1(X, Z).

The complex plane C = {z = x + iy} is oriented by

dx ∧ dy =

(

i

2

)

dz ∧ d¯ z 0.

We choose generators π1,π2 for Λ with π1 ∧ π2 0, and then the intersection

number

δ1 · δ2 = +1.

We set VZ =

H1(X,

Z), V = VZ ⊗ Q =

H1(X,

Q) and denote by

Q : V ⊗ V → Q

Q(v, v ) = −Q(v , v)

the cup-product, which via Poincar´ e duality H1(X, Q)

∼

=

H1(X, Q) is the intersec-

tion form.

We have

H1(X, C)

∼

=

HDR(X) 1 =

closed 1-forms ψ

modulo exact

1-forms ψ=dζ

∼ =

H1(X, Z)∗

⊗ C

and it may be shown that

HDR(X)

1

∼

= spanC {dz, d¯} z .

The pairing of cohomology and homology is given by periods

πi =

δi

dz

and Π =

(

π2

π1

)

is the period matrix (note the order of the πi’s).

Using the basis for

H1(X,

C) dual to the basis δ1,δ2 for H1(X, C), we have

H1(X,

C)

∼

=

C2

= column vectors

∈ ∈

dz = Π.

We may scale C by z → λz, and then Π = λΠ so that the period matrix should be

thought of as point in

P1

with homogeneous coordinates [

z0

z1

]. By scaling, we may