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474 PHILLIP A. GRIFFITHS
el
Z.Z *
where A (Ai)
e2
U(2). From
and
we deduce that
(-012
-
0) (dZl, e) A21(o]
(de1, e)
=- dA 1121 dA 1222 mod o*
F*(dL) dp / tO(A, dA)
and &*.
where tO(A, dA) is a 2-form on U(2). The integral of tO over the fibre 7r-l(z) is a
constant independent of z, and this implies (4.1).
Proof #2. This is based on the Wirtinger theorem (3.14), and the analytic
curve C will appear only at the very end.
We begin by defining the incidence correspondence
IC 2 x (1,2)
to be {(z, L) z L}. The two projections
2
7/’1
I
j 7"/" 2
(1, 2)
have respective fibres 7/’-l(z) {lines through z}, and zr-l(L) {points on the
line L}. Clearly I is a 3-dimensional complex manifold, and zr dL is a positive
(2, 2) form on I. Integration over the fibres of the first projection gives
(’W1):(’W dL)
with the properties (a)" C, i) ( is a (1, l) form on and (ii) d) is invariant under the
group of unitary motions z --) Az + b, A U(2). It follows easily that ( is a
constant multiple of the standard Kfihler form
Now setting B T/"I(C) and using (4.2) and r dLIB F* dL
IB F* dL Ic (7rl), (F* dL)
C t’ vol (C)