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.
"On the homotopy types of Käahler manifolds and the birational Kodaira problem." J. Differential Geom. 72 (1) 43 - 71, January 2006. https://doi.org/10.4310/jdg/1143593125