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 $M\ne \mathbb{CP}^1$. 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.