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].