On the classification of complex analytic supermanifolds

We consider the problem of classification of complex analytic supermanifolds with a given reduction $M$. As is well known, any such supermanifold is a deformation of its retract, i.e. of a supermanifold $(M,\mathcal O)$ whose structure sheaf $\mathcal O$ is the Grassmann algebra over the sheaf of holomorphic sections of a holomorphic vector bundle $\mathbf E\to M$. Thus, the problem is reduced to the following two classification problems: of holomorphic vector bundles over $M$ and of supermanifolds with a given retract $(M,\mathcal O$. We are dealing here with the second problem. By a well-known theorem of Green [9], it can be reduced to the calculation of the 1-cohomology set of a certain sheaf of automorphisms of $\mathcal O$. We construct a non-linear resolution of this sheaf giving rise to a non-linear cochain complex whose 1-cohomology is the desired one. For a compact manifold $M$, we apply Hodge theory to construct a finite-dimensional affine algebraic variety which can serve as a moduli variety for our classification problem; it is analogous to the Kuranishi family of complex structures on a compact manifold (see [6, 7]).