eng
episciences.org
Communications in Mathematics
1804-1388
2336-1298
2022-12-21
Volume 30 (2022), Issue 3...
10.46298/cm.9613
9613
journal article
Non-split supermanifolds associated with the cotangent bundle
Arkady Onishchik
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.
https://cm.episciences.org/9613/pdf
Mathematics - Differential Geometry
Mathematical Physics
58A50, 58A10, 32M15