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
Mots clés : homotopy of compact Kähler manifolds
@incollection{SB_2005-2006__48__55_0,
     author = {Huybrechts, Daniel},
     title = {Projectivity of K\"ahler manifolds - Kodaira's~problem},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:954},
     pages = {55--74},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     zbl = {1194.32009},
     language = {en},
     url = {http://www.numdam.org/item/SB_2005-2006__48__55_0/}
}
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://www.numdam.org/item/SB_2005-2006__48__55_0/

[1] M. Artin - Algebra, Prentice Hall Inc., Englewood Cliffs, NJ, 1991. | MR 1129886 | Zbl 0788.00001

[2] P. Deligne, P. Griffiths, J. Morgan, & D. Sullivan - “Real homotopy theory of Kähler manifolds”, Invent. Math. 29 (1975), no. 3, p. 245-274. | EuDML 142341 | MR 382702 | Zbl 0312.55011

[3] J.-P. Demailly - “Complex analytic and algebraic geometry”, http://www-fourier.ujf-grenoble.fr/~demailly/books.html.

[4] P. Griffiths & J. Harris - Principles of algebraic geometry, Pure Appl. Math., John Wiley & Sons, New York, 1978. | MR 507725 | Zbl 0836.14001

[5] H. Hironaka - “An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures”, Ann. of Math. (2) 75 (1962), p. 190-208. | MR 139182 | Zbl 0107.16001

[6] K. Kodaira - “On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)”, Ann. of Math. (2) 60 (1954), p. 28-48. | MR 68871 | Zbl 0057.14102

[7] -, “On compact analytic surfaces”, in Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, p. 121-135. | MR 140519 | Zbl 0137.17401

[8] -, “On compact complex analytic surfaces, I”, Ann. of Math. (2) 71 (1960), p. 111-152. | MR 132556 | Zbl 0098.13004

[9] K. Oguiso - “Bimeromorphic automorphism groups of non-projective hyperkähler manifolds - A note inspired by C.T. McMullen”, math.AG/0312515. | Zbl 1141.14021

[10] J. Varouchas - “Stabilité de la classe des variétés kählériennes par certains morphismes propres”, Invent. Math. 77 (1984), no. 1, p. 117-127. | EuDML 143141 | MR 751134 | Zbl 0529.53049

[11] C. Voisin - “Hodge theory and the topology of compact Kähler and complex projective manifolds”, Lecture Notes for the Seattle AMS Summer Institute.

[12] -, Théorie de Hodge et géométrie algébrique complexe, Cours spécialisés, vol. 10, Soc. Math. France, Paris, 2002. | MR 1988456 | Zbl 1032.14001

[13] -, “On the homotopy types of compact Kähler and complex projective manifolds”, Invent. Math. 157 (2004), no. 2, p. 329-343. | MR 2076925 | Zbl 1065.32010

[14] -, “On the homotopy types of Kähler manifolds and the birational Kodaira problem”, J. Differential Geom. 72 (2006), no. 1, p. 43-71. | MR 2215455 | Zbl 1102.32008