Automorphisms and real structures for a $Π$-symmetric super-Grassmannian

Abstract

Any complex-analytic vector bundle $\mathbb E$ admits naturally defined homotheties $ϕ_α$, $α\in \mathbb C^*$, i.e. $ϕ_α$ is the multiplication of a local section by a complex number $α$. 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 $Π$-symmetric super-Grassmannian $Π\!\operatorname{Gr}_{n,k}$, and, using Galois cohomology, we classify the real structures on $Π\!\operatorname{Gr}_{n,k}$ and compute the corresponding supermanifolds of real points.