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 
@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},
     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/}
}
TY  - CHAP
AU  - Huybrechts, Daniel
TI  - Projectivity of Kähler manifolds - Kodaira's problem
BT  - Séminaire Bourbaki : volume 2005/2006, exposés 952-966
AU  - Collectif
T3  - Astérisque
N1  - talk:954
PY  - 2007
SP  - 55
EP  - 74
IS  - 311
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/SB_2005-2006__48__55_0/
LA  - en
ID  - SB_2005-2006__48__55_0
ER  - 
%0 Book Section
%A Huybrechts, Daniel
%T Projectivity of Kähler manifolds - Kodaira's problem
%B Séminaire Bourbaki : volume 2005/2006, exposés 952-966
%A Collectif
%S Astérisque
%Z talk:954
%D 2007
%P 55-74
%N 311
%I Société mathématique de France
%U http://www.numdam.org/item/SB_2005-2006__48__55_0/
%G en
%F 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 | Zbl

[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 | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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