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Complex supermanifold structures being deformations of the exterior algebraof a holomorphic vector bundle, have been parametrized by orbits of a group onnon-abelian cohomology by P. Green. For the case of odd dimension $4$ and $5$an identification of these cohomologies with a subset of abelian cohomologiesbeing computable with less effort, is provided in this article. Furthermore fora 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) supermanifoldstructure associated with $\Lambda E$ to a structure associated with $\LambdaF$ is defined. In the case of $rk(F^\prime)\leq 2$ with no global derivationsincreasing the $\mathbb Z$-degree by $2$, the complete cohomologicalinformation of a supermanifold structure associated with $E$ is given in termsof cohomologies compatible with the decomposition of $E$. Details onsupermanifold structures of odd dimension 3 and 4 associated with sums of linebundles of sufficient negativity on $\mathbb P^1(\mathbb C)$ are deduced.

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