## Arkady Onishchik - Non-split supermanifolds associated with the cotangent bundle

cm:9613 - Communications in Mathematics, December 21, 2022, Volume 30 (2022), Issue 3 (Special issue: in memory of Arkady Onishchik) - https://doi.org/10.46298/cm.9613
Non-split supermanifolds associated with the cotangent bundle

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.

Volume: Volume 30 (2022), Issue 3 (Special issue: in memory of Arkady Onishchik)
Published on: December 21, 2022
Accepted on: June 22, 2022
Submitted on: May 26, 2022
Keywords: Mathematics - Differential Geometry,Mathematical Physics,58A50, 58A10, 32M15