Complex supermanifolds of odd dimension beyond 5

TL;DR

This paper studies the classification of complex supermanifolds of odd dimension beyond 5, focusing on their deformations and cohomological properties, with specific results for dimensions up to 7.

Contribution

It introduces an injection from non-abelian to abelian cohomology for certain supermanifolds and analyzes recursive construction conditions for derivations up to degree six.

Findings

Classification of supermanifolds of odd dimension up to 7.

Conditions for recursive derivation constructions.

Examples illustrating the cohomological approach.

Abstract

Any non-split complex supermanifold is a deformation of a split supermanifold. These deformations are classified by group orbits in a non-abelian cohomology. For the case of a split supermanifold with no global nilpotent even vector fields, an injection of this non-abelian cohomology into an abelian cohomology is constructed. The cochains in the non-abelian complex appear as exponentials of cochains of nilpotent even derivations. Necessary conditions for a recursive construction of these cochains of derivations are analyzed up to terms of degree six. Results on classes of examples of supermanifolds of odd dimension up to 7 are deduced.