Non-split supermanifolds associated with the cotangent bundle (Q6588450)

The main tool of the reviewed paper is a complex-analytic supermanifold in the sense of [\textit{Y. I. Manin}, Gauge field theory and complex geometry. Transl. from the Russian by N. Koblitz and J. R. King. With an appendix by S. Merkulov. 2nd ed. Berlin: Springer (1997; Zbl 0884.53002)]. More exactly, the author deals with non-split supermanifolds with retract \((M,\Omega)\), where \(\Omega=\bigoplus_{p=0}^n\Omega^p\) is the sheaf of holomorphic forms on an \(n\)-dimensional complex manifold \(M\). A construction which associates to any \(d\)-closed \((1,1)\)-form \(\omega\) on \(M\) a supermanifold with retract \((M,\Omega)\) is provided. The supermanifold obtained in this way is non-split if \(\omega\) has a non-zero Dolbeault cohomology class. To any compact Kähler manifold \(M\), the author assigns a family of supermanifolds with retract \((M,\Omega)\) parametrized by \(H^{1,1}(M,\mathbb{C})\). The only split supermanifold in this family is that corresponding to \(0\). It is shown that this family is non-empty for any flag manifold \(M\neq\mathbb{CP}^1\).\N\NAnother goal of the paper under review is the complete classification of non-split supermanifolds with retract \((M,\Omega)\), where \(M\) is a flag manifold. This goal is achieved in the case of an irreducible Hermitian symmetric space, for which the author proves that the family of non-split supermanifolds with retract \((M,\Omega)\) has exactly one element if one excludes the Grassmannians \(M = {\mathrm{Gr}}^n_s\), where \(2\leq s\leq n-2\), whose non-split supermanifolds form an \(1\)-parameter family.\N\NFor all the supermanifolds assigned to the cotangent bundle over a compact irreducible Hermitian symmetric space, the \(0\)-cohomology and \(1\)-cohomology of the tangent sheaf are computed.\N\NFinally, the author deals with the supermanifolds of \(\Pi\)-symmetric flags constructed by Yu. Manin, i.e. the \(\Pi\)-symmetric super-Grassmannians \(\Pi{\mathrm{Gr}}^{n|s}_{n|s}\). More explicitly, the author studies the necessary and sufficient conditions for splitness, homogeneity and rigidity of the super-Grassmannians \(\Pi{\mathrm{Gr}}^{n|s}_{n|s}\).