## A.L.Onishchik On the classification of complex analytic supermanifolds (Lobachevskii Journal of Mathematics, Vol.4, pp.47-70)

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 $\M$ whose structure sheaf $\Cal O$ is the Grassmann algebra over the sheaf of holomorphic sections of a holomorphic vector bundle $\bold E\to M$. Thus, the problem is reduced to the following two classification problems: of holomorphic vector bundles over $M$
and of supermanifolds with a given retract $\M$. We are dealing here with the second problem. By a well-known theorem of Green, it can be reduced to the calculation of the 1-cohomology set of a certain sheaf of automorphisms of $\Cal O$. 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.

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