Homogeneous supermanifolds over Grassmannians (Q2370308)

If one considers a complex flag manifold \(M=G/P\) where \(G\) is a complex semisimple Lie group and \(P\) its parabolic group, the author studies the problem of describing (up to isomorphy) all homogeneous complex supermanifolds \((M, \mathcal {O})\) which have \(M\) as their reduction. He considers the case where \(M\) is a complex Grassmannian manifold \(Gr_{n,k}\) and under certain restrictions on the representation \(\phi\) of the subgroup \(P\) which determines the retract of \((M, \mathcal {O})\). Thus, in the split case, the solution is given for all completely reducible representations. In the non-split case, \(\phi\) is supposed to be irreducible and only one non-split homogeneous supermanifold exists for the cases: \(3\leq k\leq n-k\), \(k=2\) and \(n\) odd and \(n\geq 5\) and for \(Gr_{4,2}\). The same result for \(Gr_{6,2}\) is announced. Preliminaries on supermanifolds are also included in order to facilitate the comprehension of the results.