Geodesic
Compact complex homogeneous spaces with solvable fundamental group
Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 61-83.

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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. 
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D. N. Akhiezer. Compact complex homogeneous spaces with solvable fundamental group. Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 61-83. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a4/

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