Locally free sheaves on complex supermanifolds

A. L. Onishchik; E. G. Vishnyakova

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arXiv · 1110.3908

Locally free sheaves on complex supermanifolds

A. L. Onishchik·E. G. Vishnyakova

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Abstract

An important part of the classical theory of real or complex manifolds is the theory of (smooth, real analytic or complex analytic) vector bundles. With any vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or complex analytic) sections is associated which is a locally free sheaf of F-modules, and in this way all the locally free sheaves of F-modules over (M,F) can be obtained. In the present paper, locally free sheaves of O-modules over a complex analytic supermanifold (M,O) are studied. The main results of the paper are the following ones. Given a locally free sheaf E of O-modules over a complex analytic supermanifold (M,O), we construct a locally free sheaf over the retract of (M,O) which is called the retract of E. Our first result is a classification of locally free sheaves of modules which have a given retract in terms of non-abelian 1-cohomology. The case of the tangent sheaf of a split supermanifold is studied in more details. Then we study locally free sheaves of modules over projective superspaces. A spectral sequence which connects the cohomology with values in a locally free sheaf of modules with the cohomology with values in its retract is constructed.

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BibTeX RIS

A. L. Onishchik, E. G. Vishnyakova. 2011-12-05. Locally free sheaves on complex supermanifolds. https://arxiv.org/abs/1110.3908

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