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
