Dolbeault cohomology of compact complex homogeneous manifolds

We show that if M is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group of M acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spaces W, which we call generalized Hopf manifolds, which are diffeomorphic to S1×K/L where K is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus in K. We compute the Dolbeault cohomology of W. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group