On the homotopy types of kähler manifolds and the birational kodaira problem

Abstract

In the paper [8], 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 equiva- lent to complex projective manifolds. Thus it remained possible that in higher dimension, a birational version of Kodaira’s theo- rem, 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. © 2006 Journal of Differential Geometry.