On the holomorphic reduction of pseudoconvex complex homogeneous manifolds

I will explain a generalization of a recent result of Kim, Levenberg and Yamaguchi concerning the obstruction for a pseudoconvex domain spread over a complex homogeneous manifold $X=G/H$ to be Stein. This will then be applied to study the holomorphic reduction of $X=G/H$ under the assumptions that $X$ itself is pseudoconvex and that $G$ is solvable or reductive. These results have been obtained jointly with Bruce Gilligan and Karl Oeljeklaus.