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 $(M,\Omega )$, 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.

arXiv comment79 pages

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