Normal almost contact structures and non-Kaehler compact complex manifolds

Abstract

We construct some families of complex structures on compact manifolds by means of normal almost contact structures (nacs) so that each complex manifold in the family has a non-singular holomorphic flow. These families include as particular cases the Hopf and Calabi-Eckmann manifolds and the complex structures on the product of two normal almost contact manifolds constructed by Morimoto. We prove that every compact Kaehler manifold admitting a non-vanishing holomorphic vector field belongs to one of these families and is a complexificacion of a normal almost contact manifold. Finally we show that if a complex manifold obtained by our constructions is Kaehlerian the Euler class of the nacs (a cohomological invariant associated to the structure) is zero. Under extra hypothesis we give necessary and sufficient conditions for the complex manifolds so obtained to be Kaehlerian.