Projectivity of Kähler manifolds - Kodaira's problem
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 954, pp. 55-74.

Every compact Kähler surface is deformation equivalent to a projective surface. In particular, topologically Kähler surfaces and projective surfaces cannot be distinguished. Kodaira had asked whether this continues to hold in higher dimensions. We explain the construction of a series of counter-examples due to C. Voisin, which yields compact Kähler manifolds of dimension at least four whose rational homotopy type is not realized by any projective manifold.

Toute surface kählérienne compacte est déformation d’une surface projective. En particulier, topologiquement il n’y a pas de différence entre surfaces kählériennes et surfaces projectives. Kodaira avait demandé si ceci reste vrai en dimension supérieure. On expliquera la construction d’une série de contre-exemples dus à C. Voisin, qui construit des variétés kählériennes compactes de dimension ≥4 dont le type d’homotopie rationnelle ne peut être celui d’une variété projective.

Classification: 32J27, 14F35, 32J25
Keywords: homotopie des variétés kählériennes compactes
Mot clés : homotopy of compact Kähler manifolds 
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Huybrechts, Daniel. Projectivity of Kähler manifolds - Kodaira's problem, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 954, pp. 55-74. http://archive.numdam.org/item/SB_2005-2006__48__55_0/

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