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490 PHILLIP A. GRIFFITHS For this we decompose into its tangential part and normal part Z Z "4r Zn Zt Z a,e, Zn E z.e.. We may consider zn as a holomorphic section of the line bundle Q in (4.35), and the curvature matrix for this line bundle is the (1, 1) form (4.36) q -0c] log iiz.ii Setting z, zn we obtain a unitary frame {el, ", e,; z.} for F. The Chern form c(O) may be computed using this frame, as well as the previous one {Z0, ", Z,}. The curvature matrix of F in the first frame will be denoted by To compute it we use the second fundamental form of T(M) in F--the point is that we know the curvature matrix of T(M) and qY of Q, and want to determine from these and the second fundamental form the curvature matrix of F. Summing repeated indices the second fundamental form of T(M) C F is given by the vector of (1, 0) forms co,,. (de,, - It follows that 12.. co,. A d,. (to check signs, recall that curvatures decrease on sub-bundles and increase on quotient bundles). There are similar formulas for 1],. and 12.,. From (4.37) we infer that (4.38) cI(OM) Cl(’M) + IM
CURVATURE AND COMPLEX SINGULARITIES 491 Using this we will prove (4.28) when n k 1; i.e., when the intersections M f3 L0 have complex dimension one. Now (4.38) is valid for any complex manifold in (I u {0}, and applying it to M f3 L0 and integrating gives IM (4.39) f) Lo Cl(M f) L) fM f-) Lo cI(OM fq L) fM f- Lo M f3 Lo" We now average both sides of (4.39) over Lo G(N- n + 1, N) and use (4.32) to obtain I (4.40) ( fM c io Cl(UCi)) dL fM cI(OM) / ("on-1 + Cm fu on I( IU o Xlru c I) dL" We must examine the term on the far right. In s {0} x G(N n + l, N) we consider the incidence correspondence I= {z, L0) :z L0} (actually, I should be considered in IP N X G(N n + 1, N)). The fibre of rr: I-- [u {0} is Iz G(N n, N 1), and so rr,(dLo) is an (n 1, n 1) form on C u {0}. Since this form is the pullback of a form on pu- and is unitarily invariant, it is a multiple of o" 1. We may then write (4.41) dLo where is a form on I which restricts to the fundamental class on each fibre Iz. Here we are using that the cohomology of I is additively isomorphic to H,(IpN- 1) (R) H*(G(N n, N 1)), and in this decomposition the cohomology class of dLo appears in H 2n 1) (ipN- 1) () H*(G(N n, N 1)). Now we denote by z,,(Lo) the normal vector for M f3 L0 C L0 at z. Since Tz(M f) Lo) T(M) fq Lo it follows by an easy invariant-theoretic argument that log IlzdLo)l[ Combining this with (4.41) and (4.40) gives c’- log Ilznll. O log IIz.(Lo)ll cte IM --0 log iiz.ii ,--1
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Vol. 45, No. 3 DUKE MATHEMATICAL JO
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CURVATURE AND COMPLEX SINGULARITIES
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we deduce that CURVATURE AND COMPLE
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Page 79 and 80: As a consequence of (5.15), CURVATU
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