  We recently constructed examples of compact Kaeler 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 still holds. We
construct however in this paper compact Kaehler manifolds, no birational model
of which has the homotopy type of a projective manifold.
