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

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.