Compact complex homogeneous spaces with solvable fundamental group

In this paper, complex Lie groups $G$ acting transitively and effectively on complex manifolds $X$ with solvable (nilpotent) fundamental groups are studied. It is shown that if $\pi_1(X)$ is nilpotent, then locally $G=S\times N$, where $S$ is semisimple and $N$ is nilpotent. In the case when $\pi_1(X)$ is solvable, the Levi decomposition of the group $G$ is direct if and only if the stationary subgroup contains a maximal unipotent subgroup of the semisimple part. The question of the existence of transitive semisimple groups on $X$ is considered.