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CURVATURE AND COMPLEX SINGULARITIES 469
I m tflt C(m, k) + 9k
Ikll r
where rn N- n. Indeed, the unitary group U(m) acts irreducibly on the
space Symk((Em*) of homogeneous polynomials of degree k in tn + 1, ", tN; it
leaves invariant the inner product
I
(t, t)=
where do- is the invariant volume on the sphere, as well as the inner product
induced from the standard one on (m. It follows from Schur’s lemma that these
two inner products are proportional. Setting
we deduce that
where
Pk(’M)- cte
cte
dM A %/--1 (to, A tb,)
i Pk(M)rm + 2 dM
k=O
since h,u
C ,g, R,:,I,I... R,,
by (3.9). Observe that this step is easier than in the real case where skewsymmetry
conditions intervene. More importantly, using
we deduce that
where
4)=
2
P(tq)dM C(tqM) A 49"-
Ck(’M) cte Z A
is, by (3.5), the kth basic Chern polynomial in the curvature matrix f of M.
Summarizing, we have arrived at our desired result:
(3.18) vol rr(M) C(k, n, m)/z(m+ k) Ck("M) i 6 n- k.
k=O