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CURVATURE AND COMPLEX SINGULARITIES 473
where dL is a suitably normalized invariant measure on (1, 2).
In fact, we will give three quite different proofs of (4.1). The first will be a
direct computation using frames analogous to the proof of (2.1). The remaining
two will be based on general invariant-theoretic principles, and will generalize
to give proofs of our two main integral-geometric formulas below.
Proof #1. Associated to C is the manifold if(C) of Darboux frames
{z*; e, e} where z* C and e Tz,(C) (cf. the discussion in 3(a)). The
structure equations are
dz* Olel * *
de le + *lze*z
We observe that and are forms of type (1, 0), and that writing h
the Khler form (= volume form) and st given respectively by
Chern form in the tangent bundle are
-1
2
A
c(Oc)
-1
2
4 K’
w ]z A &]z
where K -41hi is the Gaussian curvature of the Riemann surface C.
Associated to a line L (1, 2) are the frames {z; e, e} where L is the line
through z in the direction el. Recalling the structure equations (3.1), we infer as
h the real case that
dL C o2 A 52 A 6012 A (.12.
Proceeding as we did there, set
---
B {(z, L):z L} C (E
so that we have a diagram
B 0(1,2)
C
x ((1, 2)
The left hand side of (4.1) is [ F* dL, and we shall evaluate this integral by
integration over the fibres of r. Fixing z C, the lines through z may be given
parametrically by