Homogeneous supermanifolds over Grassmannians

The problem of classifying all homogeneous complex supermanifolds whose reduction is the complex Grassmannian Grn,k is studied. Under the assumption that the odd isotropy representation is irreducible and under certain restrictions on (n,k), we prove that the only non-split supermanifold of this sort is the Π-symmetric super-Grassmannian constructed by Yu.I. Manin.