  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.
