12 1. THE CLASSICAL THEORY: PART I

Riemann surface of genus g 2 with the metric induced from that on H. By the

Gauss-Bonnet theorem

0 2 − 2g = χ(Y ) =

1

4π

Y

KdA = K

Area(Y )

4π

.

Proof of basic calculation. We define a section s ∈ Γ(H, V1,0) by

s(τ) =

τ

1

∈

Vτ,0.1

The length squared is given by

s(τ)

2

=

its(τ)Qs(τ)

= 2y.

Using for τ = x + iy

∂τ =

1

2

(∂x − i∂y)

∂¯=

τ

1

2

(∂x + i∂y)

we obtain

i

2π

∂∂ = −

1

4π

(∂x

2

+ ∂y

2)dx

∧ dy.

This gives

i

2π

∂∂ log s(τ)

2

=

1

4π

dx ∧ dy

y2

.

Remark. There is also a SU(2)-invariant metric on OP1 (−1) induced from the

standard metric on

C2.

For this metric

s(τ)

2

c

= 1 +

|τ|2

(the subscript c on

2

c

stands for “compact”). Then we have

c1(Vc,0) 1

= −

1

4π

dx ∧ dy

(1 + |τ|2)2

.

Thus, V1,0 → H is a positive line bundle whereas Vc,0 1 → P1 is a negative line

bundle with

deg OP1 (−1) =

P1

c1(V1,0)

c

= −1.

This sign reversal between the SL2(R)-invariant curvature on the open domain H

and the SU(2) (= compact form of SL2(C))-invariant metric on the compact dual

ˇ

H =

P1

will hold in general and is a fundamental phenomenon in Hodge theory.

Above we have holomorphically trivialized

V1,0

→ H using the section

s(τ) =

τ

1

.

We have also noted that we have the isomorphism of SL2(R)-homogeneous line

bundles

ωH

∼

=

V2,0.

Now ωH has a section dτ and a useful fact is that under this isomorphism

dτ =

s(τ)2.