##
*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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