Locally free sheaves on complex supermanifolds

A classification of locally free sheaves $ \mathcal{E} $ of $ \mathcal{O} $ -modules which have a given retract gr $ \mathcal{E} $ in the terms of non-abelian 1-cohomology is given. In the case of $ \mathbb{C}{{\mathbb{P}}^{1|m }} $ , m > 0, we show that the Birkhoff–Grothendieck Theorem does not hold true. We obtain a result similar to the Barth–Van de Ven–Tyurin Theorem for projective superspaces. Furthermore, a spectral sequence which connects the cohomology with values in a locally free sheaf $ \mathcal{E} $ to the cohomology with values in its retract gr $ \mathcal{E} $ is constructed.