Automorphisms and real structures for a $\Pi$-symmetric super-Grassmannian

Abstract

Any complex-analytic vector bundle $\mathbb{E}$ admits naturally defined homotheties ${\varphi }_{\alpha }$, $\alpha \in {\mathbb{C}}^{\ast }$, i.e. ${\varphi }_{\alpha }$ is the multiplication of a local section by a complex number $\alpha$. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to $\mathbb{E}$. Further, we compute the automorphism supergroup of a $\Pi$-symmetric super-Grassmannian $\Pi {\mathrm{Gr}}_{n,k}$, and, using Galois cohomology, we classify the real structures on $\Pi {\mathrm{Gr}}_{n,k}$ and compute the corresponding supermanifolds of real points.

42 Pages, appendix written by Mikhail Borovoi. A missed case fixed, the paper has been significantly revised

Cited by in corpus (1)

• Non-split supermanifolds associated with the cotangent bundle