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488 PHILLIP A. GRIFFITHS n J where L 7r(L0) varies over the Grassmannian of IP N k 1’ s in IP N 1o We may view (4.31) as a local averaged version of the usual adjunction formulas in algebraic geometry. (ii) Retaining the preceding notations, we consider the vector bundle F whose fibre over 2o is the (n + 1)-plane spanned by Zo, ", Z,. The associated projective bundle is the bundle of tangent projective spaces to/f/c IP N 1, and we have the Euler sequence where H* /f/is the universal line bundle with fibres Z0, and T(/) is the holomorphic tangent bundle. The connection matrix for the Hermitian connection in F is 0. (dz., z), and the curvature matrix is From 00o 0 it follows that Using (4.31) we will prove that 0 dO 0 A 0 0ao A Ooo 0 0o., and lO"=- OoAOo. -- O kavoooOo (4.32) Cn- k(Oa n r) dL Ct Cl(OM) / on- 1. nL =o Proof. According to (4.30) the relation between the curvature matrices =andO=Ois (4.33) We will show that 0oz 0o + z 0 + Oz A 0o. k (4.34) ck(O) A o" k Cl(c(O) A oo"- ), Ct 1, /=0 which when combined with (4.31) will establish (4.32).
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(.
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Vol. 45, No. 3 DUKE MATHEMATICAL JO
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CURVATURE AND COMPLEX SINGULARITIES
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Page 11 and 12: 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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