Codimension one holomorphic foliations with trivial canonical class

Frédéric Touzet

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 4, page 657-670
  • ISSN: 0012-9593

Abstract

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Nous décrivons les variétés kählériennes compactes M de dimension complexe n dont le fibré tangent admet un sous-fibré holomorphe intégrable ℱ de rang n - 1 dont le fibré déterminant D é t ℱ est à première classe de Chern nulle. Ce résultat peut en quelque sorte être considéré comme un avatar feuilleté du théorème de Bogomolov concernant les variétés kählériennes à fibré canonique numériquement trivial.

How to cite

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Touzet, Frédéric. "Feuilletages holomorphes de codimension un dont la classe canonique est triviale." Annales scientifiques de l'École Normale Supérieure 41.4 (2008): 657-670. <http://eudml.org/doc/272162>.

@article{Touzet2008,
abstract = {We give a description of Kähler manifolds $M$ equipped with an integrable subbundle $\mathcal \{F\}$ of $TM$ of rank $n-1$ ($n=\mathrm \{dim\}\ M$) under the assumption that the line bundle $\mathrm \{Dét\}\ \{\mathcal \{F\}\}$ is numerically trivial. This is a sort of foliated version of Bogomolov’s theorem concerning Kähler manifolds with trivial canonical class.},
author = {Touzet, Frédéric},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {holomorphic foliatios; canonical class of a codimension 1 foliation; compact Kähler manifolds},
language = {fre},
number = {4},
pages = {657-670},
publisher = {Société mathématique de France},
title = {Feuilletages holomorphes de codimension un dont la classe canonique est triviale},
url = {http://eudml.org/doc/272162},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Touzet, Frédéric
TI - Feuilletages holomorphes de codimension un dont la classe canonique est triviale
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 4
SP - 657
EP - 670
AB - We give a description of Kähler manifolds $M$ equipped with an integrable subbundle $\mathcal {F}$ of $TM$ of rank $n-1$ ($n=\mathrm {dim}\ M$) under the assumption that the line bundle $\mathrm {Dét}\ {\mathcal {F}}$ is numerically trivial. This is a sort of foliated version of Bogomolov’s theorem concerning Kähler manifolds with trivial canonical class.
LA - fre
KW - holomorphic foliatios; canonical class of a codimension 1 foliation; compact Kähler manifolds
UR - http://eudml.org/doc/272162
ER -

References

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  1. [1] A. Beauville, Variétés kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom.18 (1983), 755–782. Zbl0537.53056MR730926
  2. [2] F. A. Bogomolov, Kähler manifolds with trivial canonical class, Izv. Akad. Nauk SSSR Ser. Mat.38 (1974), 11–21. Zbl0292.32020MR338459
  3. [3] M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. École Norm. Sup.30 (1997), 569–594. Zbl0893.32019MR1474805
  4. [4] M. Brunella, Plurisubharmonic variation of the leafwise Poincaré metric, Internat. J. Math.14 (2003), 139–151. Zbl1052.32027MR1966769
  5. [5] M. Brunella, A positivity property for foliations on compact Kähler manifolds, Internat. J. Math.17 (2006), 35–43. Zbl1097.37041MR2204838
  6. [6] M. Brunella, J. V. Pereira & F. Touzet, Kähler manifolds with split tangent bundle, Bull. Soc. Math. France134 (2006), 241–252. Zbl1187.32018
  7. [7] J. Cheeger & D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom. 6 (1971/72), 119–128. Zbl0223.53033
  8. [8] M. Kellum, Uniformly quasi-isometric foliations, Ergodic Theory Dynam. Systems13 (1993), 101–122. Zbl0784.58060MR1213081
  9. [9] S. Kobayashi & K. Nomizu, Foundations of differential geometry. Vol. II, John Wiley & Sons Inc., 1996. Zbl0175.48504
  10. [10] S. Kobayashi & H. Wu, Complex differential geometry, DMV Seminar 3, Birkhäuser, 1987. 
  11. [11] F. Loray & J. C. Rebelo, Minimal, rigid foliations by curves on ℂ ℙ n , J. Eur. Math. Soc.5 (2003), 147–201. Zbl1021.37030
  12. [12] J. Martinet & J.-P. Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. École Norm. Sup.16 (1983), 571–621. Zbl0534.34011
  13. [13] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer, 1972. Zbl0254.22005MR507234
  14. [14] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), 339–411. Zbl0369.53059MR480350

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