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

dx dy
y2
=
i


(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 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
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