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460 PHILLIP A. , GRIFFITHS dz o, e Setting do, 0 in the first equation of (3.1) gives o A o O, which again by the Cartan lemma implies that The first and second fundamental forms of M C N are defined by I o(R)o3 The other part of the first equation in (3.1) is (3.8) &o, ’, oe A oe, oe + &an O. It is well known that given a K/ihler metric I there is a unique matrix {o} of 1forms satisfying (3.8), which is then the connection matrix for the canonical Hermitian connection in the holomorphic tangent bundle T(M). The curvature matrix "M {"o/3} is Setting it follows that
CURVATURE AND COMPLEX SINGULARITIES 461 As in the real case we may consider the holomorphic Gauss mapping y M----) G(n,N) with y*E T(M). The Hermitian connection and curvature are induced from those in E G(n, N). There is a commutative diagram m. G(n, N) G(2n, 2N) where YR is the usual real Gauss mapping, and we deduce from (3.6) and the discussion at the end of c that the Gauss Bonnet integrand is y*Pf((P) y*c,(l)) c,(1); i.e., (3.10) C e KdM C,(M). NOW a natural question is whether the Gaussian image y(M) has dimension n, and using (3.7) we shall prove: For S C (" + a complex-analytic surface (3.11) c2(1s) >- 0 with equality holding only if (i) S is a plane, (ii) S is a developable ruled surface; (4) or (iii) S is a cone. Proof. We shall adhere to the notations in the proof of (3.7). If c2(1t) 0 and alternative ii) holds, then we may choose a local moving frame {el, e; ea, ., e, + .} so that From de1 (.Ollel, and thus 0.112 (121 0, de2 (.022e2 + (.o23e3 + (.o24e4. (023 h213(.01 -+- h223(.02 and we deduce that (.023 h223(.02 Similarly (.024 h224(.02, and it follows that the Gaussian image T(S) is a curve. The fibres of T:S 7(S) are defined by o2 0; these are curves along which dz (.o el, del 0911 el. We infer that these fibres are straight lines. If alternative (ii) in (3.7) holds, then we may choose a moving flame so that
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Page 11 and 12: we deduce that CURVATURE AND COMPLE
Page 13 and 14: so that CURVATURE AND COMPLEX SINGU
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Page 41 and 42: 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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CURVATURE AND COMPLEX SINGULARITIES
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