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

TL;DR

This paper studies automorphisms and real structures of $ ext{Pi}$-symmetric super-Grassmannians, analyzing when automorphisms lift to supermanifolds and classifying real structures using Galois cohomology.

Contribution

It computes the automorphism supergroup of $ ext{Pi}$-symmetric super-Grassmannians and classifies their real structures, providing new insights into their automorphisms and real forms.

Findings

Automorphism supergroup of $ ext{Pi}$-symmetric super-Grassmannian computed.

Conditions for lifting automorphisms to supermanifolds established.

Classification of real structures on $ ext{Pi}$-symmetric super-Grassmannians achieved.

Abstract

Any complex-analytic vector bundle $\mathbb E$ admits naturally defined homotheties $\phi_{\alpha}$, $\alpha\in \mathbb C^*$, i.e. $\phi_{\alpha}$ 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 $\Pi\!\operatorname{Gr}_{n,k}$, and, using Galois cohomology, we classify the real structures on $\Pi\!\operatorname{Gr}_{n,k}$ and compute the corresponding supermanifolds of real points.