A note on compact solvmanifolds with Kähler structures

We know that the existence of Kähler structure on a compact complex manifold imposes certain homological or even homotopical restrictions on its underlining topological manifold. Hodge theory is of central importance in this line. There have been recently certain extensions and progresses in this area of research. Among them is the field of Kähler groups, in which the main subject to study is the fundamental group of a compact Kähler manifold. Once there was a conjecture that a non-abelian, finitely generated and torsion-free nilpotent group (which is the fundamental group of a nilmanifold) can not be a Kähler group, which is a generalized assertion of the result ([7, 12]) that a non-toral nilmanifold admits no Kähler structures. A counter example to this conjecture was given by Campana [10]. Later a detailed study of solvable Kähler groups was done by Arapura and Nori [2]; they showed in particular that a solvable Kähler group must be almost nilpotent, that is, it has a nilpotent subgroup of finite index. On the other hand, the author stated in the paper [13] a general conjecture on compact Kählerian solvmanifolds: a compact solvmanifold admits a Kähler structure if and only if it is a finite quotient of a complex torus which is also a complex torus bundle over a complex torus; and showed under some restriction that the conjecture is valid.