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478 PHILLIP A. GRIFFITHS
is zero. But by the third property, for g U(N)
g*q q
and, as in the previous argument, a is an invariant current of odd degree and is
therefore identically zero. Q.E.D.
As an example, suppose we consider the sequence (k, 0, ., 0); the corresponding
Schubert cycle will be denoted by Xk. For k n it is determined by a
hyperplane H IPu 1,, and is described by
(4.10) 2;, {T G(n, N); T C/.
In general, the polynomial corresponding to the Schubert cycle Xk is
(- 1) c(fl) where c(12) is the kth basic Chern form given explicitly by (3.5).
As an application of (4.9), we consider a complex manifold M C (I2 N and consider
its image under the holomorphic Gauss mapping r
We recall from section 3(a) that
y" M---> G(n, N).
T(M)
fM
y’E; and
and therefore (4.9) implies that
IM (4.11) Pa(M) (y(M), Xa) d ,a
f#
for any codimension n Schubert cycle Xa. In particular, taking a
infer from (4.10)
n(H) we
(4.12) # (y(M), Xn(H))
number of points z M where the tangent
plane Tz(M) lies in z + H
n(M, H)
where the last equality is a definition. Since by (3.10)
(--1) n Cn(nm) C te K dM
is the Gauss-Bonnet integrand, we have arrived at what might be called the
average Gauss-Bonnet theorem
(4.13) (-1),cte IM KdM In n(M, H) dH
where the integrand on the right is given by (4.12).
To interpret this equation we recall the usual form
(4.14) cte IM KdM=x(M) + IOM kg