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Uniformisation de l’espace des feuilles de certains feuilletages de codimension un

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Résumé

Dans cet article, nous étudions, sur des variétés Kähler compactes, les feuilletages holomorphes (éventuellement singuliers) dont le fibré conormal est pseudoeffectif. En utilisant la notion de courant à singularités minimales, nous montrons que l’on peut munir canoniquement l’espace des feuilles d’une métrique à courbure constante négative ou nulle dont les éventuelles dégénerescences sont localisées le long d’une hypersurface invariante “rigidement plongée” dans la variété.

Abstract

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a “rigidly” embedded invariant hypersurface.

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Références

  1. L. Alessandrini and G. Bassanelli. Plurisubharmonic currents and their extension across analytic subsets. Forum Math., t. 5 (1993), pp. 577–602.

    MathSciNet  MATH  Google Scholar 

  2. S. Bochner and W.T. Martin. Several Complex variables. Princeton University press, (1948).

    MATH  Google Scholar 

  3. S. Boucksom. Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4), 37(1) (2004), 45–76.

    MathSciNet  MATH  Google Scholar 

  4. M. Brunella. Birational geometry of foliations. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2004).

    Google Scholar 

  5. M. Brunella. Feuilletages holomorphes sur les surfaces complexes compactes. Ann. Sci. École Norm. Sup. (4), 30(5) (1997), 569–594.

    MathSciNet  MATH  Google Scholar 

  6. M. Brunella. Courbes entières et feuilletages holomorphes. Enseign. Math. (2), 45(1-2) (1999), 195–216.

    MathSciNet  MATH  Google Scholar 

  7. M. Brunella, J.V. Pereira and F. Touzet. Kähler manifolds with split tangent bundle. Bull. Soc. Math. France, 134(2) (2006), 241–252.

    MathSciNet  MATH  Google Scholar 

  8. D. Cerveau and J.-F. Mattei. Formes intégrables holomorphes singulières. Astérisque, 97 Société Mathématique de France, Paris (1982), 193 pp.

    Google Scholar 

  9. S.D. Cutkosky. Zariski decomposition of divisors on algebraic varieties. Duke Math. J., 53 (1986), 149–156.

    Article  MathSciNet  MATH  Google Scholar 

  10. J.P. Demailly. On the Frobenius integrability of certain holomorphic p-forms. Complex geometry (Göttingen, 2000), 93–98, Springer, Berlin (2002).

    Chapter  Google Scholar 

  11. J.P. Demailly. L 2 vanishing theorems for positive line bundles and adjunction theory. Transcendental methods in algebraic geometry. Lectures given at the 3rd C.I.M.E. Session held in Cetraro, July 4–12, (1994). Edited by F. Catanese and C. Ciliberto. Lecture Notes in Mathematics, 1646 (1-97).

  12. C. Godbillon. Feuilletages. Études géométriques. Progress in Mathematics, 98. Birkhäuser Verlag, Basel (1991).

    MATH  Google Scholar 

  13. P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York (1994), 813 p.

    MATH  Google Scholar 

  14. L. Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin (2003).

    MATH  Google Scholar 

  15. M. Klimek. Pluripotential theory. London Mathematical Society Monographs. New Series, 6. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991).

    MATH  Google Scholar 

  16. F. Loray. Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Prépublication IRMAR (2005).

    Google Scholar 

  17. F. Loray, J.V. Pereira and F. Touzet. Singular foliations with trivial canonical class, arXiv:1107.1538, (2011), 68 pages.

    Google Scholar 

  18. J.-F. Mattei and R. Moussu. Holonomie et intégrales premières. Ann. Sc. ENS, 13 (1980), 469–523.

    MathSciNet  MATH  Google Scholar 

  19. L.G. Mendes and J.V. Pereira. Hilbert modular foliations on the projective plane. Comment. Math. Helv., 80(2) (2005), 243–291.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Molino. Riemannian foliations. Translated from the French by Grant Cairns. With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu. Progress in Mathematics, 73. Birkhäuser Boston, Inc., Boston, MA (1988), 339 pp.

    Book  MATH  Google Scholar 

  21. E. Paul. Feuilletages holomorphes singuliers à holonomie résoluble. J. Reine Angew. Math., 514 (1999), 9–70.

    MathSciNet  MATH  Google Scholar 

  22. E. Paul. Connectedness of the fibers of a Liouvillian function. Publ. Res. Inst. Math. Sci., 33(3) (1997), 465–481.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Reid. Bogomolov’s theorem c 21 ≤ 4c 2. Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), pp. 623–642, Kinokuniya Book Store, Tokyo (1978).

    Google Scholar 

  24. B.L. Reinhart. Foliated manifolds with bundle-like metrics. Ann. of Math. (2), 69 (1959), 119–132.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Seidenberg. Reduction of singularities of the differential equation Ady = Bdx. Amer. J. Math., 89 (1968), 248–269.

    Article  MathSciNet  Google Scholar 

  26. N. Shepherd-Barron. Miyaoka’s theorems on the generic seminegativity of TX and on the kodaira dimension of minimal regular threefolds. J. Kollár et al., “Flips and abundance for algebraic threefolds”, Astérisque, 211 (1993).

  27. H. Skoda. Prolongement des courants, positifs, fermés de masse finie. Invent. Math., 66(3) (1982), 361–376.

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Touzet. Feuilletages holomorphes de codimension 1 dont la classe canonique est triviale. Ann. Sci. École Norm. Sup., 41 (2008), 1–14.

    MathSciNet  Google Scholar 

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  1. IRMAR, Campus de Beaulieu, 35042, Rennes Cedex, France

    Frédéric Touzet

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  1. Frédéric Touzet
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Correspondence to Frédéric Touzet.

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Touzet, F. Uniformisation de l’espace des feuilles de certains feuilletages de codimension un. Bull Braz Math Soc, New Series 44, 351–391 (2013). https://doi.org/10.1007/s00574-013-0017-7

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