## Journal of Differential Geometry

### On the homotopy types of Käahler manifolds and the birational Kodaira problem

Claire Voisin

#### Abstract

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.

#### Article information

Source
J. Differential Geom., Volume 72, Number 1 (2006), 43-71.

Dates
First available in Project Euclid: 28 March 2006

https://projecteuclid.org/euclid.jdg/1143593125

Digital Object Identifier
doi:10.4310/jdg/1143593125

Mathematical Reviews number (MathSciNet)
MR2215455

Zentralblatt MATH identifier
1102.32008

#### Citation

Voisin, Claire. On the homotopy types of Käahler manifolds and the birational Kodaira problem. J. Differential Geom. 72 (2006), no. 1, 43--71. doi:10.4310/jdg/1143593125. https://projecteuclid.org/euclid.jdg/1143593125