A classification theorem and a spectral sequence for a locally free sheaf cohomology of a supermanifold

TL;DR

This paper introduces a new filtration, classification theorem, and spectral sequence for locally free sheaves on supermanifolds, providing a more convenient approach in certain cases, expanding the theoretical framework of supermanifold sheaf cohomology.

Contribution

It presents an alternative filtration, classification theorem, and spectral sequence for locally free sheaves on supermanifolds, differing from previous work and offering practical advantages.

Findings

Developed a new filtration for locally free sheaves

Established a classification theorem based on this filtration

Constructed a spectral sequence that is more convenient in some cases

Abstract

This paper is based on the paper "Locally free sheaves on complex supermanifolds" of A.L.Onishchik, E.G. Vishnyakova, where two classification theorems for locally free sheaves on supermanifolds were proved and a spectral sequence for a locally free sheaf of modules E was obtained. We consider another filtration of the locally free sheaf E, the corresponding classification theorem and the spectral sequence, which is more convenient in some cases. The methods, which we are using here, are similar to the methods of the paper mentioned above. The first spectral sequence of this kind was constructed by A.L. Onishchik for the tangent sheaf of a supermanifold. However, the spectral sequence considered in this paper is not a generalization of Onishchik's spectral sequence.