Non-split supermanifolds associated with the cotangent bundle

Abstract

Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold (M, Ω), where Ω is the sheaf of holomorphic forms on a given complex manifold M of dimension > 1. I propose a general construction associating with any d-closed (1, 1)-form ω on M a supermanifold with retract (M, Ω) which is non-split whenever the Dolbeault class of ω is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold M = CP 1 . In the case where M is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract (M, Ω). For each of these supermanifolds, the 0-and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the Π-symmetric super-Grassmannians introduced by Yu. Manin.