8 1. THE CLASSICAL THEORY: PART I
Definitions. (i) A Hodge structure of weight one is given by a Q-vector space
V with a line V
1,0
⊂ VC satisfying
VC = V
1,0
⊕ V
0,1
V
0,1=
V
1,0
.
(ii) A polarized Hodge structure of weight one (PHS) is given by the above together
with a non-degenerate form
Q : V ⊗ V → Q, Q(v, v ) = −Q(v , v)
satisfying the Hodge-Riemann bilinear relations
Q(V
1,0,V 1,0)
= 0
iQ(V
1,0,
V
1,0
) 0.
In practice we will usually have V = VZ ⊗ Q. The reason for working with Q
will be explained later.
When dim V = 2, we may always choose a basis so that V
∼
=
Q2
= column
vectors and Q is given by the matrix above. Then V
1,0
∼
= C is spanned by a point
[
τ
1
] ∈ PVC
∼
=
P1
Identification. The space of PHS’s of weight one (period domain) is given by
the upper-half-plane
H = {τ : Im τ 0}.
The compact dual
ˇ
H given by subspaces V
1,0
⊂ VC satisfying Q(V
1,0,V 1,0)
= 0
(this is automatic in this case) is
ˇ
H = PVC
∼
=
P1 where
P1
= {τ-plane} ∪ ∞ = lines through the origin in
C2.2
It is well known that H and P1 are homogeneous complex manifolds; i.e., they
are acted on transitively by Lie groups. Here are the relevant groups. Writing
z =
z0
z1
, w =
w0
w1
and using Q to identify Λ2V with Q we have
Q(z, w) =
twQz
= z ∧ w
and the relevant groups are
Aut(VR,Q)
∼
=
SL2(R) for H
Aut(VC,Q)
∼
= SL2(C) for
P1.
In terms of the coordinate τ the action is the familiar one:
τ →
aτ + c
cτ + d
where
(
a b
c d
)
∈ SL2. This is because τ = z0/z1 and
a b
c d
z0
z1
=
az0 + bz1
cz0 + dz1
= z1
aτ + b
cτ + d
.
2[CM-SP]
is a general reference for period domains and their differential geometric proper-
ties. A recent source is [Ca].
