Flat partial connections and holomorphic structures in vector bundles

Author:
J. H. Rawnsley

Journal:
Proc. Amer. Math. Soc. **73** (1979), 391-397

MSC:
Primary 58A30; Secondary 32L10, 58F06

DOI:
https://doi.org/10.1090/S0002-9939-1979-0518527-X

MathSciNet review:
518527

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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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0518527-X

Keywords:
Flat partial connections,
holomorphic vector bundles,
fine resolution,
polarization

Article copyright:
© Copyright 1979
American Mathematical Society