Projectivity of Kähler manifolds - Kodaira's problem
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 954, p. 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
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     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},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:954},
     pages = {55-74},
     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, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk 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. | 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 0408.14001 | 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. | 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