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468 PHILLIP A. GRIFFITHS Kfihler form on (I N pulled back to N[r] becomes b= /-1 ( The volume form on ([N is N We shall iterate the volume integral on N[r]; thus setting we find (z, r) j Itll O(z, t) vol rr(M) I qt(z, r). To evaluate the right hand side we fix z, assume as we may that our moving frame has been chosen so that o.(z) 0 (this is convenient but not essential), and set dt dtn + A A diN o o, A Aro. h(t). h(t)Ao- k! det(h(t),) where A (o1, o, Ok) and D (31, ", 3k) are index sets (1, ., n). Denoting by A the index set complementary to A, selected from A 49. +- h(t)AOOAO A A,D 13 B,C where it is understood that the summation is over index sets having the same number of elements. It follows that For/x (/zl, ",/xk) with/xl
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
Page 1 and 2: Vol. 45, No. 3 DUKE MATHEMATICAL JO
Page 3 and 4: CURVATURE AND COMPLEX SINGULARITIES
Page 11 and 12: we deduce that CURVATURE AND COMPLE
Page 13 and 14: so that CURVATURE AND COMPLEX SINGU
Page 41: while by Stokes’ theorem CURVATUR
Page 45 and 46: we conclude that CURVATURE AND COMP
Page 63 and 64: For k we have (4.33) CURVATURE AND
Page 79 and 80: As a consequence of (5.15), CURVATU

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