Complex supermanifolds of low odd dimension and the example of the complex projective line

TL;DR

This paper studies complex supermanifolds of low odd dimension, providing cohomological classifications and reductions, with specific examples on the complex projective line, enhancing understanding of their structure and parametrization.

Contribution

It offers a simplified cohomological approach for classifying supermanifolds of odd dimensions 4 and 5 and introduces a reduction method for structures associated with subbundles.

Findings

Cohomological classification for odd dimensions 4 and 5.

Reduction of supermanifold structures to subbundles.

Explicit analysis of supermanifolds on b^1 with line bundles.

Abstract

Complex supermanifold structures being deformations of the exterior algebra of a holomorphic vector bundle, have been parametrized by orbits of a group on non-abelian cohomology by P. Green. For the case of odd dimension $4$ and $5$ an identification of these cohomologies with a subset of abelian cohomologies being computable with less effort, is provided in this article. Furthermore for a rank $\leq 3$ sub vector bundle $F\to M$ of a holomorphic vector bundle $E=F\oplus F^\prime\to M$, a reduction of a (possibly non-split) supermanifold structure associated with $\Lambda E$ to a structure associated with $\Lambda F$ is defined. In the case of $rk(F^\prime)\leq 2$ with no global derivations increasing the $\mathbb Z$-degree by $2$, the complete cohomological information of a supermanifold structure associated with $E$ is given in terms of cohomologies compatible with the decomposition of $E$. Details on supermanifold structures of odd dimension 3 and 4 associated with sums of line bundles of sufficient negativity on $\mathbb P^1(\mathbb C)$ are deduced.