Non existence de structures Kaehleriennes sur le fibres principaux en tores

Abstract

The main results of this paper concerns principal circle bundles P with non vanishing Euler class, over odd-dimensional, closed, oriented and connected bases B. Such manifolds do not admit any Kaehler structure. We show that the same holds for principal torus bundles whose characteristic classes span a subgroup of, at least, rank one in the second integral cohomology group of the base space. This approach provides a new proof of a theorem of Benson and Gordon which asserts that no nilmanifold but the torus can be made kaehlerian. We end this note by extending this last result to certain classes of compact solvmanifolds.