Projectivity of Kähler manifolds – Kodaira’s problem

Daniel Huybrechts

Séminaire Bourbaki (2005-2006)

  • Volume: 48, page 55-74
  • ISSN: 0303-1179

Abstract

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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.

How to cite

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Huybrechts, Daniel. "Projectivity of Kähler manifolds – Kodaira’s problem." Séminaire Bourbaki 48 (2005-2006): 55-74. <http://eudml.org/doc/252176>.

@article{Huybrechts2005-2006,
abstract = {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.},
author = {Huybrechts, Daniel},
journal = {Séminaire Bourbaki},
keywords = {homotopie des variétés kählériennes compactes},
language = {eng},
pages = {55-74},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Projectivity of Kähler manifolds – Kodaira’s problem},
url = {http://eudml.org/doc/252176},
volume = {48},
year = {2005-2006},
}

TY - JOUR
AU - Huybrechts, Daniel
TI - Projectivity of Kähler manifolds – Kodaira’s problem
JO - Séminaire Bourbaki
PY - 2005-2006
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 48
SP - 55
EP - 74
AB - 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.
LA - eng
KW - homotopie des variétés kählériennes compactes
UR - http://eudml.org/doc/252176
ER -

References

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  9. [9] K. Oguiso – “Bimeromorphic automorphism groups of non-projective hyperkähler manifolds – A note inspired by C.T. McMullen”, math.AG/0312515. Zbl1141.14021
  10. [10] J. Varouchas – “Stabilité de la classe des variétés kählériennes par certains morphismes propres”, Invent. Math. 77 (1984), no. 1, p. 117–127. Zbl0529.53049MR751134
  11. [11] C. Voisin – “Hodge theory and the topology of compact Kähler and complex projective manifolds”, Lecture Notes for the Seattle AMS Summer Institute. 
  12. [12] —, Théorie de Hodge et géométrie algébrique complexe, Cours spécialisés, vol. 10, Soc. Math. France, Paris, 2002. Zbl1032.14001MR1988456
  13. [13] —, “On the homotopy types of compact Kähler and complex projective manifolds”, Invent. Math. 157 (2004), no. 2, p. 329–343. Zbl1065.32010MR2076925
  14. [14] —, “On the homotopy types of Kähler manifolds and the birational Kodaira problem”, J. Differential Geom. 72 (2006), no. 1, p. 43–71. Zbl1102.32008MR2215455

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