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Let $G$ be a connected complex Lie group, $H$ a closed, complex subgroup of $G$ and $X:=G/H$. Let $R$ be the radical and $S$ a maximal semisimple subgroup of $G$. Attempts to construct examples of noncompact manifolds $X$ homogeneous under a nontrivial semidirect product $G=S⋉R$ with a not necessarily $G$-invariant Kähler metric motivated this paper. The $S$-orbit $S/S\cap H$ in $X$ is Kähler. Thus $S\cap H$ is an algebraic subgroup of $S$ [4]. The Kähler assumption on $X$ ought to imply the $S$-action on the base $Y$ of any homogeneous fibration $X\to Y$ is algebraic...

We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.