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

Y
KdA = K
Area(Y )

.
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

∂∂ =
1

(∂x
2
+ ∂y
2)dx
dy.
This gives
i

∂∂ log s(τ)
2
=
1

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

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 and a useful fact is that under this isomorphism
=
s(τ)2.
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