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502 PHILLIP A. GRIFFITHS
representing a codimension n Schubert condition on G(n, N) and denote by
Pa(fE) the corresponding polynomial in the curvature representing the cohomology
class dual to the fundamental cycle of Za. Then, from 3(b)
IVo
lirn Pa(E)=
On the other hand, for -
0
Pa(V) fVt Pa(vt)
where D, vt is the curvature matrix in the tangent bundle of Vt, while by the
discussion of 3(c) for 0 the integral
converges and is equal to
o(o) V*
From this together with (5.8) we infer that
P(n)"
lim I Pa(ft)= I Pa(f0)+ I Pa().
t-O
We may evaluate the term on the far right by the first Crofton formula (4.9) and
obtain
(5.10)
01im fvt Pa(vt)= fVo Pa(v) + f#
t---
(A, a)da
This is our main general result expressing the difference between the limit of
the curvature and the curvature of the limit. It is clear that there is an analogous
result for a family of either analytic or algebraic varieties in IPN.
In case Vo has only isolated interior singularities, A is an algebraic variety in
G(n, N), the intersection number # (A, a) is constant for all Schubert cycles
in the family, and (5.10) becomes our
Main Formula (I). For a family of complex-analytic varieties Vt acquiring
an isolated singularity
(5.11) t-01im fvt Pa(vt) IVo Pa(v) + # (A, Xa)
where A is the Pliicker defect. Note that if we combine (5.11) with the Crofton
formula (4.9) we obtain
(5.12) t---01im f #((Vt) a)da f # (y(V0), a)da + # (A, a)