10.46298/cm.9613
https://cm.episciences.org/9613
Onishchik, Arkady
Arkady
Onishchik
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.
Comment: 79 pages
episciences.org
Mathematics - Differential Geometry
Mathematical Physics
58A50, 58A10, 32M15
Attribution 4.0 International (CC BY 4.0)
2022-06-22
2022-12-21
2022-12-21
eng
journal article
arXiv:2205.12308
10.48550/arXiv.2205.12308
2336-1298
https://cm.episciences.org/9613/pdf
VoR
application/pdf
Communications in Mathematics
Volume 30 (2022), Issue 3 (Special issue: in memory of Arkady Onishchik)
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