Download Complex Differential Geometry: Topics in Complex by Shoshichi Kobayashi, Camilla Horst, Hung-Hsi Wu (auth.) PDF
By Shoshichi Kobayashi, Camilla Horst, Hung-Hsi Wu (auth.)
Read or Download Complex Differential Geometry: Topics in Complex Differential Geometry Function Theory on Noncompact Kähler Manifolds PDF
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Extra info for Complex Differential Geometry: Topics in Complex Differential Geometry Function Theory on Noncompact Kähler Manifolds
A global section Tp q := ® 9 ® ®g 1 • Locally,~ is of the form p q (as usual we omit the tensor symbol). Denote by g~ the metric on T~ induced by g = g~ (see 1 • 3. 5) , and by V~ the corresponding hermitian connection, and let gP(~ ® ~) is a real-valued c=-function on M and q locally IIV ~11 2 + Q(~). v 'X. V azK = -g <'r
5. 1. 6 If r = o, then every holomorphic tensor field on M is parallel. 5. 1. 7 If r ~ o, then every holomorphic covariant tensor field is parallel; if r::: o, then every holomorphic contravariant tensor field is parallel. 5. 1. 8 If r ~ o and r (x) > o for some x E M, then =0 (i) (ii) H0 (M,T~) = o If r::: o and r (x) < o (i) H0 (M, ® 9) p =0 (ii) H0 (M,T~) = o for all q ~ 1 , and for q » p. for some x E M, then for all p for p » q. e. 9 (i) = o for all q>p, and every holomorphic tensor field of covariant and contravariant degree p is parallel; (ii) if r < o, then H0 (M,T~) = o for all q < p, and every holomorphic tensor field of covariant and contravariant degree p is parallel.
7 Thus for a compact M to admit a conformal structure it is necessary that 2c 1 be divisible by n (in H 2 (M,~)). 8 For a compact surface the above condition is of course always fulfilled. Ce T(p 1 ,o)M ~ - ~ in two linearly independent lines. Thus there exists a two-fold unramified covering n : M' ~ M such that T( 1 ,o)M' ~ n*T( 1 ,o)M is the direct sum of two line bundles. 3 Let M be compact with a holomorphic conformal metric g. 1) in T( 1 ' 0 )M whose local curvature matrices are of the form 9 1 + 9 2 , 9 1 of type (1,1), 9 2 of type (2,o) such that det (I+ v-r 2rt.