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CURVATURE AND COMPLEX SINGULARITIES 491
Using this we will prove (4.28) when n k 1; i.e., when the intersections
M f3 L0 have complex dimension one.
Now (4.38) is valid for any complex manifold in (I u {0}, and applying it to
M f3 L0 and integrating gives
IM (4.39)
f) Lo Cl(M f) L) fM f-) Lo cI(OM fq L) fM f- Lo M f3 Lo"
We now average both sides of (4.39) over Lo G(N- n + 1, N) and use
(4.32) to obtain
I (4.40) ( fM c io Cl(UCi)) dL fM cI(OM) / ("on-1
+ Cm fu on I( IU o Xlru c I) dL"
We must examine the term on the far right.
In s {0} x G(N n + l, N) we consider the incidence correspondence
I= {z, L0) :z L0}
(actually, I should be considered in IP N X G(N n + 1, N)). The fibre of
rr: I-- [u {0} is Iz G(N n, N 1), and so rr,(dLo) is an (n 1, n 1)
form on C u {0}. Since this form is the pullback of a form on pu- and is
unitarily invariant, it is a multiple of o" 1. We may then write
(4.41) dLo
where is a form on I which restricts to the fundamental class on each fibre Iz.
Here we are using that the cohomology of I is additively isomorphic to
H,(IpN- 1) (R) H*(G(N n, N 1)), and in this decomposition the cohomology
class of dLo appears in H 2n 1) (ipN- 1) () H*(G(N n, N 1)). Now we denote
by z,,(Lo) the normal vector for M f3 L0 C L0 at z. Since
Tz(M f) Lo) T(M) fq Lo
it follows by an easy invariant-theoretic argument that
log IlzdLo)l[
Combining this with (4.41) and (4.40) gives
c’- log Ilznll.
O log IIz.(Lo)ll
cte IM --0 log iiz.ii ,--1