On the classification of complex analytic supermanifolds
This paper deals with the problem of classifying complex analytic (Berezin-Kostant) supermanifolds with a given associated split supermanifold \((M,\bigwedge \mathcal E)\), where \(M\) is a complex manifold and \(\mathcal E\) a holomorphic vector bundle. When \(E\) is the sheaf of holomorphic 1-forms, the problem was solved by the author [\textit{A. L. Onishchik}, Proceedings of the colloquium, Tashkent, Uzbekistan, September 29--October 5, 1997, 23-24 (1998; Zbl 0931.32004)]. The work is based on \textit{P. Green}'s fundamental result [Proc. Am. Math. Soc. 85, 587-590 (1982; Zbl 0509.58001)] that there is a bijection between the set of supermanifolds having \((M,\bigwedge \mathcal E)\) as their retract and the quotient of the first cohomology group of a certain sheaf of automorphisms of the graded algebra \(\bigwedge \mathcal E\) modulo the natural action of the automorphisms of the bundle \(\mathcal E\). This cohomology group can be computed in terms of the supertangent bundle, as was shown by \textit{H. J. Rothstein} [Proc. Am. Math. Soc. 95, 255-260 (1985; Zbl 0584.32041)]. In this paper an explicit nonlinear resolution of the supertangent bundle is given, thus allowing another proof of Rothstein's result. Moreover, new applications to Hodge theory are deduced when \(M\) is compact and hermitian metrics are being fixed on both \(M\) and \(\mathcal E\). The author constructs an affine algebraic variety which plays in this situation the role of the Kuranishi family of complex structures on a compact manifold.
scientific article; zbMATH DE number 1413874