Flat partial connections and holomorphic structures in 𝐶^{∞} vector bundles

Rawnsley, J. H.

doi:10.1090/S0002-9939-1979-0518527-X

Flat partial connections and holomorphic structures in $C^{\infty }$ vector bundles
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Proc. Amer. Math. Soc. 73 (1979), 391-397 Request permission

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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