Locally free sheaves on complex supermanifolds

TL;DR

This paper extends classical vector bundle theory to complex supermanifolds, classifying locally free sheaves of modules and exploring their cohomological properties, including applications to tangent sheaves and projective superspaces.

Contribution

It introduces a classification of locally free sheaves over supermanifolds via non-abelian 1-cohomology and constructs a spectral sequence linking their cohomologies.

Findings

Constructed a sheaf over the retract of a supermanifold from a given sheaf.

Classified locally free sheaves with a fixed retract using non-abelian 1-cohomology.

Developed a spectral sequence relating cohomology of sheaves and their retracts.

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.