Projectivity of Kähler manifolds - Kodaira's problem
[Projectivité des variétés kählériennes - le problème de Kodaira]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 954, pp. 55-74.

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.

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.

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

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