  It is well-known that non-constant holomorphic functions do not exist on a
compact complex manifold. This statement is false for a supermanifold with a
compact reduction. In this paper we study the question under what conditions
non-constant holomorphic functions do not exist on a 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 which were introduced by Yu.I. Manin.
