A. L. Onishchik, E. G. Vishnyakova. Locally free sheaves on complex supermanifolds
Submitted on: Apr 11, 2012, 06:55:17
Natural Sciences / Mathematics / Differential equations
Description: An important part of the classical theory of real or complex manifolds is the theory of 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. 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. 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.
The abstract of this article has been published in the "Intellectual Archive Bulletin" , April 2012, ISSN 1929-1329.
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