Flat Partial Connections end Holomorphic Structures in C^∞ Introduction Vector Bundles

Rawnsley, J. H. (1978) Flat Partial Connections end Holomorphic Structures in C^∞ Introduction Vector Bundles. (Preprint)

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Abstract

The notion of a flat partial connection D in a C^∞ vector bundle E, defined on an integrable sub-bundle 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^∞ 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 characterised as extensions of the representation of the isomorphic subgroup to which E is associated, extending results of Tirao and Wolf in the holomorphic case.

Item Type:	Article
Divisions:	School of Theoretical Physics > Preprints
Date Deposited:	11 Jul 2018 10:53
Last Modified:	16 Dec 2022 00:12
URI:	https://dair.dias.ie/id/eprint/957

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