Non-split supermanifolds associated with the cotangent bundle

TL;DR

This paper classifies non-split supermanifolds associated with the cotangent bundle of complex manifolds, providing a construction based on closed (1,1)-forms and analyzing their properties, especially on Hermitian symmetric spaces.

Contribution

It introduces a general construction linking closed (1,1)-forms to non-split supermanifolds and offers a complete classification for certain complex manifolds.

Findings

Constructs non-split supermanifolds from (1,1)-forms with non-zero Dolbeault class

Provides a classification for supermanifolds over Hermitian symmetric spaces

Calculates cohomology groups for these supermanifolds

Abstract

Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,\Omega)$, where $\Omega$ 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 $\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-split whenever the Dolbeault class of $\omega$ 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 $(M,\Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.