Normal almost contact structures and non-K\"ahler 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 Kahler manifold admitting a non-vanishing holomorphic vector field belongs to one of these families, and is a complexification of a normal almost contact manifold. Finally, we show that if a complex manifold obtained by our constructions is Kahlerian, then the Euler class of the nacs (a cohomological invariant associated to the structure) is zero. Under extra hypotheses, we give necessary and sufficient conditions for the complex manifolds so obtained to be Kahlerian.