By Shoshichi Kobayashi, Camilla Horst, Hung-Hsi Wu (auth.)

ISBN-10: 303486566X

ISBN-13: 9783034865661

ISBN-10: 376431494X

ISBN-13: 9783764314941

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Extra info for Complex Differential Geometry: Topics in Complex Differential Geometry Function Theory on Noncompact Kähler Manifolds

Example text

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. dz~ with ga = faBgB on uanuB. 6 we may assume (after multiplying g with an appropriate o-cochain in O*) that on ua, and 50 n ua. Then, just as in the Riemannian case, we get on ua i azb azc Ai - a zl3 a a a cal3jk cabc ] I C az~ azl3 azl3 i a zl3 a 2 z~ j k az~ azaazl3 ' and thus i az~ az~ - azl3 ca i cl3jk abc aza azj JC a a aza i azl3 a2za a 1 j k - 2fal3 az~ azaaza (o~ afal3 afaa - ik afal3) --k- + oi ga gl3jk k azj azh a zl3 13 13 i In other words, the o-cochain (Cajk e ® a az~ S21 ® S21 is a solution of the cocycle := .

1 {i} If c 1 ~ o, then h o {M,S m52}_ m , and where d 0 denotes the dimension of the Euclidean factor of any de Rham decomposition of M. 2 if M has finite fundamental group, and if c 1 = o, then there are no non-zero holomorphic covariant or contravariant symmetric tensor fields on M. For instance, 5. 5. 3 K 3 - surfaces admit no non - zero holomorphic covariant or contravariant symmetric tensor fields. 4 If c 1 O for some m > o, then M is covered by a product D x N, where D is a positive-dimensional hermitian symmetric space.

2 the (ga) define a global section g in 48 ~1 0n ® ~1 °n ® 0 JPn+1 (2)1Q n • Actually, g€H 0 (Qn,s 2 ~6 ®OJP (2) 1 0 ), as Q is synunetric. n n+2 n n Furthermore, for any local coordinates (z 1 , •. e. g= g .. dzidzj) is nowhere ~J degenerate. Thus g defines an isomorphism e0 n 1 ~ ~On(2). 4 Up to multiplication with a o-cochain in O* g is invariant under the action of G. Therefore, if M carries a quadric structure, then pulling back g via the quadric charts one obtains a section g€ H 0 (M,s 2 ~~ ®F), F some line bundle on M, which induces an isomorphism eM 1 ~ ~M ® F.

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Complex Differential Geometry: Topics in Complex Differential Geometry Function Theory on Noncompact Kähler Manifolds by Shoshichi Kobayashi, Camilla Horst, Hung-Hsi Wu (auth.)


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