On the classification of complex analytic supermanifolds
(Lobachevskii Journal of Mathematics, Volume IV)
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 is the Grassmann algebra over the sheaf of holomorphic sections of a holomorphic vector bundle E→ 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 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]).