On the homotopy types of Kahler manifolds and the birational Kodaira problem

In the paper [8], we constructed examples of compact Kahler manifolds which do not have the homotopy type of a projective complex manifold. They were, however, obtained by blowing-up certain complex tori, which axe themselves deformation equivalent to complex projective manifolds. Thus it remained possible that in higher dimension, a birational version of Kodaira's theorem, saying that a compact Kahler surface deforms to a projective surface, still holds. We construct in this paper compact Kahler manifolds, no smooth birational model of which, however, has the homotopy type of a projective manifold. Thus the possibility mentioned above is excluded, even at the topological level.