{"docId":9613,"paperId":9613,"url":"https:\/\/cm.episciences.org\/9613","doi":"10.46298\/cm.9613","journalName":"Communications in Mathematics","issn":"1804-1388","eissn":"2336-1298","volume":[{"vid":631,"name":"Volume 30 (2022), Issue 3 (Special issue: in memory of Arkady Onishchik)"}],"section":[],"repositoryName":"arXiv","repositoryIdentifier":"2205.12308","repositoryVersion":2,"repositoryLink":"https:\/\/arxiv.org\/abs\/2205.12308v2","dateSubmitted":"2022-05-26 07:42:55","dateAccepted":"2022-06-22 22:03:33","datePublished":"2022-12-21 12:11:53","titles":["Non-split supermanifolds associated with the cotangent bundle"],"authors":["Onishchik, Arkady"],"abstracts":["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.","Comment: 79 pages"],"keywords":["Mathematics - Differential Geometry","Mathematical Physics","58A50, 58A10, 32M15"]}