Flat partial connections and holomorphic structures in vector bundles
J. H. Rawnsley
Proc. Amer. Math. Soc. 73 (1979), 391-397
Primary 58A30; Secondary 32L10, 58F06
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Abstract: The notion of a flat partial connection D in a 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 . The sheaf of germs of sections s of E satisfying 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 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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