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For k we have (4.33)
CURVATURE AND COMPLEX SINGULARITIES 489
c() (n + 1)0 + c1(O)
which trivially implies (4.34). For k 2, which is the crucial case, we have
from (4.33)
c2()/ to"- c2(O)/ to"- + Cto,, + L(O)to,
where L(O) is linear in the entries of O. From (4.30) we deduce that L(O) is a
linear combination of the expressions
where repeated indices will now be summed. From the theory of unitary invariants
(cf. the reference in footnote ) of 4) we infer that L(O) is a linear
combination of the two expressions
B the smmetr k k these are both equal, and since
2 0 2
. k
L(O)" Cc(O) A "-
(kvokOov A 0o).
This establishes (4.34) for k 2, and the general argument is similar. We note
again the essential role played by the symmetry of the second fundamental
form.
We also remark that, by the discussion in 3(c), the formula (4.32) extends to
the case where c N- may have singularities; the point is that both sides
are defined as the corresponding integrals over smooth points and these integrals
are absolutely convergent.
(iii) Now let M C N {0} be a complex manifold with residual image in
N- 1. Over M we consider the usual Darboux frames {z, e, ., eN}, and over
we have the frames {Z0,’’ ", ZN-} () where Z0 ez/ z]]. Over
smooth points of we have z A e A A e, 0 and the two sets of vectors
{z, e, ., e,} and {Z0, ", Z,} both span the fibre of the bundle F at Z. The
pair of exact sequences
(4.35)
0H*F T()H*0
O T(M) F Q O
contains the relationship between the bundles T(M) and T(), both of which
are complex-analytically isomorphic to the holomorphic tangent bundle of M
but which have quite different metrics. By (4.32) we have averaging formulas
for the Chern forms c(O) of F, and we want to use these to deduce averaging
formulas for the Chern forms c() of T(.