On holomorphic functions on a compact complex homogeneous supermanifold

It is well known that non-constant holomorphic functions do not exist on a connected compact complex manifold. This statement is false for a supermanifold with a connected compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a connected compact homogeneous complex supermanifold. We describe also the vector bundles determined by split homogeneous complex supermanifolds. As an application, we compute the algebra of holomorphic functions on the classical flag supermanifolds introduced in Manin (1997) [11].