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48O PHILLIP A. GRIFFITHS
Grassmannian of oriented affine (N- k) planes,
(4.18) f ( L
I(R L)) dL Ct-e I I(RM)
where the It(R) are the curvature polynomials (1.9) appearing in Hermann
Weyl’s tube formula (1.10). In the complex case we have seen in (3.18) that the
integrals in the tube formula are just those on the right hand side of (4.17), and
consequently this result is the complex analogue of Chern’s formula (4.18).
Now presumably the real proof could be adapted to give (4.17). However, it
is convenient to take advantage of properties peculiar to the complex case, and
so we shall give an argument along the lines of the second proof of (4.1). The
following notations and ranges of indices will be used:
G(l, N) Grassmann manifold of ’s through the origin in CN;
((l, N) Grassmannian of affine/-planes in (IN;
0(1, N) flag variety of pairs {z, L} ([U X ((l, N) where z L; note that
(l, N) (N G(l ,N)
where {z, L} maps to (z, L-z) with L_ denoting the translate of L
by -z;
(l, m, N) set of triples {z, S, L} CN dT(l ,N) 0(m, N) where
zSCL;
U--> ((1, m, N)is the universal vector bundle with fibre S_ over {z, S, L};
and
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