1. THE CLASSICAL THEORY: PART I 11

where

Vz1,0

is the line in VC corresponding to z. If we use the group SL2(C) that

preserves Q in place of GL2(C), then

VC/Vz1,0

∼

=

Vz1,0∗

.

Thus the cotangent space

Tz

∗ P1

∼

=

Vz2,0

where in general we set

Vn,0

=

(V1,0)⊗n.

The above identification ωP1

∼

=

OP1 (−2)

is an SL2(C), but not GL2(C), equivalence of homogenous bundles.

Convention. We set

ωH

1/2

=

V1,0.

The Hodge bundle

V1,0

→ H has an SL2(R)-invariant metric, the Hodge metric,

given fibrewise by the

2nd

Hodge-Riemann bilinear relation. The basic invariant of

a metric is its curvature, and we have the following

General fact. Let L → Y be an Hermitian line bundle over a complex mani-

fold Y . Then the Chern (or curvature) form is

c1(L) =

1

2πi

∂∂ log s

2

where s ∈ O(L) is any non-vanishing local holomorphic section and s 2 is its length

squared.

Basic calculation.

c1(V1,0)

=

1

4π

dx ∧ dy

y2

=

i

2π

dτ ∧ dτ

(Im τ)2

.

This has the following

Consequence. The tangent bundle

T H

∼

=

V0,2

has a metric

dsH

2

=

dx2

+

dy2

y2

=

1

(Im τ)2

Re(dz d¯) z

of constant negative Gauss curvature.

Before giving the proof we shall make a couple of observations.

Any SL2(R) invariant Hermitian metric on H is conformally equivalent to

dx2

+

dy2;

hence it is of the form

h(x, y)

dx2 + dy2

y2

for a positive function h(x, y). Invariance under translation τ → τ + b, b ∈ R,

corresponding to the subgroup ( 1 b

0 1

), implies that h(x, y) = h(y) depends only on

y. Then invariance under τ → aτ corresponding to the subgroup

a1/2

0

0

a−1/2

,

a 0, gives that h(y) = constant. A similar argument gives that

c1(V1,0)

is a

constant multiple of the form above.

The all important sign of the curvature K may be determined geometrically

as follows: Let Γ ⊂ SL2(R) be a discrete group such that Y = Γ\H is a compact