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

Any complex-analytic vector bundle E admits naturally defined homotheties ϕ_α, α∈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 E. Further, we compute the automorphism supergroup of a Π-symmetric super-Grassmannian ΠGr_n,k, and, using Galois cohomology, we classify the real structures on ΠGr_n,k and compute the corresponding supermanifolds of real points.