Two families of flag supermanifolds (Q1345398)

The authors construct two one-parametric analytic families of compact complex supermanifolds admitting a transitive action of the family of exceptional Lie superalgebras \(D(2,1;\alpha)\). The isotropy subalgebras of these actions are parabolic. The first family contains the projective superline \(\mathbb{C} \mathbb{P}^{1| 4}\), and the second one the super- Grassmannian \(G_{2| 2, 1| 1}\). The tangent sheaf cohomology of the constructed supermanifolds is computed.