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Flat partial connections and holomorphic structures in $ C\sp{\infty }$ vector bundles

Author: J. H. Rawnsley
Journal: Proc. Amer. Math. Soc. 73 (1979), 391-397
MSC: Primary 58A30; Secondary 32L10, 58F06
MathSciNet review: 518527
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Abstract: The notion of a flat partial connection D in a $ {C^\infty }$ vector bundle E, defined on an integrable subbundle F of the complexified tangent bundle of a manifold X is defined. It is shown that E can be trivialized by local sections s satisfying $ Ds = 0$. The sheaf of germs of sections s of E satisfying $ Ds = 0$ has a natural fine resolution, giving the de Rham and Dolbeault resolutions as special cases.

If X is a complex manifold and F the tangents of type (0, 1), the flat partial connections in a $ {C^\infty }$ vector bundle E are put in correspondence with the holomorphic structures in E.

If X, E are homogeneous and F invariant, then invariant flat connections in E can be characterized as extensions of the representation of the isotropic subgroup to which E is associated, extending results of Tirao and Wolf in the holomorphic case.

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Keywords: Flat partial connections, holomorphic vector bundles, fine resolution, polarization
Article copyright: © Copyright 1979 American Mathematical Society

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