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

Permanent link to this document
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.


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