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CURVATURE AND COMPLEX SINGULARITIES 477
to the flag F in the amount specified by (4.7). The Schubert cycle forms an
irreducible analytic variety of complex codimension lal a in the Grassmannian,
and clearly
g Z.(F) X.(gF), g U(N).
Up to a constant there is a unique invariant measure on the space of Schubert
cycles with fixed index sequence a; for simplicity of notation we shall write a
general Schubert cycle as E with d denoting the invariant measure.
We recall from section 3(a) the curvature matrix f (fo) of the universal
n-plane bundle E G(n, N). A fundamental result (5) is that the de Rham cohomology
of the Grassmannian is represented by the invariant polynomials
P(12) in the curvature matrix, and moreover these are just the polynomials in
the basic Chern forms ce(f) defined by (3.5). In particular, the fundamental
class of each Schubert cycle Ea(F) is represented by a polynomial
which clearly does not depend on the particular choice of flag. For any closed
algebraic subvariety V c G(n, N) of dimension [a the intersection relation
Iv (4.8) Pa(f) # (V,
is valid. We wish to prove that (4.8) is also true locally on the average. More
precisely, we will prove that for V c G(n, N) a piece of analytic variety of pure
dimension d [a I, we have Crofton’s formula (I):
Iv (4.9) Pa(’E)= (V, ,a) f#
Proof. This is formally the same as our third proof of (4.1). For each flag
F F(N) there is a (2d 1) form rF,a which satisfies, (n)
i) "OF, is integrable on G(n, N) and C on the complement of
ii) the equation of currents
d’OF,a Pa(I"E) TXa(F)
is valid; and
iii) for any unitary transformation g U(N),
gF,a g*TIF,a"
The formula (4.9) is of a local character and so we may assume V is a complex
manifold with smooth boundary. Then for any F F(N) such that a(F) meets
V at a finite number of interior points, by property (ii) and Stokes’ theorem
fV Pa(’E) # (V, Xa(F)) + Iov TF’a"
Averaging this formula over U(N), we are reduced to showing that the average