On a special class of fibrations and Kähler rigidity (Q816312)

It is proved that for any exact sequence \(1 \to N \to \Gamma \to \Phi \to 1\), where \(N\) is torsion-free, finitely generated maximal nilpotent and \(\Phi\) is a finite group, there is a compact Kähler \(K (\Gamma,1)\)-manifold \(M\) if and only if \(N\cong {\mathbb Z}^{2n}\) and the operator homomorphism \(\varphi : \Phi \to \Aut N\) is essentially complex. From this result it is derived that any torsion-free, virtually polycyclic group \(\Gamma\) can be realized as the fundamental group of a \(K(\pi,1)\) compact Kähler manifold if and only if the group \(\Gamma\) is a Bieberbach group with essentially complex operator homomorphism.