Previously, we constructed examples of compact Kähler
manifolds which do not have the homotopy type of a projective
complex manifold. They were, however, obtained by blowing-up
certain complex tori, which are 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 Kähler surface deforms to a projective
surface, still holds. We construct in this paper compact Kähler 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.