Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Rationally connected foliations after Bogomolov and McQuillan

Authors: Stefan Kebekus, Luis Solá Conde and Matei Toma
Journal: J. Algebraic Geom. 16 (2007), 65-81
Published electronically: May 25, 2006
MathSciNet review: 2257320
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Abstract | References | Additional Information

Abstract: This paper is concerned with sufficient criteria to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov and McQuillan and using the recent work of Graber, Harris, and Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic $ p$.

We also give a new sufficient condition to ensure that all leaves are algebraic.

The results are then applied to show that $ \mathbb{Q}$-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration.

References [Enhancements On Off] (What's this?)

  • [BM01] Feodor A. Bogomolov and Michael L. McQuillan.
    Rational curves on foliated varieties.
    IHES, Preprint, February 2001.
  • [Bos01] Jean-Benoît Bost.
    Algebraic leaves of algebraic foliations over number fields.
    Publ. Math. Inst. Hautes Études Sci., 93:161-221, 2001. MR 1863738 (2002h:14037)
  • [CLN85] César Camacho and Alcides Lins Neto.
    Geometric theory of foliations.
    Birkhäuser Boston Inc., Boston, MA, 1985.
    Translated from the Portuguese by Sue E. Goodman. MR 0824240 (87a:57029)
  • [Deb01] Olivier Debarre.
    Higher-dimensional algebraic geometry.
    Universitext. Springer-Verlag, New York, 2001. MR 1841091 (2002g:14001)
  • [Fle84] Hubert Flenner.
    Restrictions of semistable bundles on projective varieties.
    Comment. Math. Helv., 59(4):635-650, 1984. MR 0780080 (86m:14014)
  • [GHS03] Tom Graber, Joe Harris, and Jason Starr.
    Families of rationally connected varieties.
    J. Amer. Math. Soc., 16(1):57-67 (electronic), 2003. MR 1937199 (2003m:14081)
  • [Gro66] Alexandre Grothendieck.
    Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III.
    Inst. Hautes Études Sci. Publ. Math., 28:255, 1966. MR 0217086 (36:178)
  • [Gro95] Alexandre Grothendieck.
    Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert.
    In Séminaire Bourbaki, Vol. 6, pages Exp. No. 221, 249-276. Soc. Math. France, Paris, 1995. MR 0217086 (36:178)
  • [Har68] Robin Hartshorne.
    Cohomological dimension of algebraic varieties.
    Ann. of Math., 88(2):403-450, 1968. MR 0232780 (38:1103)
  • [Har71] Robin Hartshorne.
    Ample vector bundles on curves.
    Nagoya Math. J., 43:73-89, 1971. MR 0292847 (45:1929)
  • [Hör05] Andreas Höring. Uniruled varieties with splitting tangent bundle. preprint math.AG/0505327, 2005.
  • [Kol91] János Kollár.
    Extremal rays on smooth threefolds.
    Ann. Sci. École Norm. Sup. (4), 24(3):339-361, 1991. MR 1100994 (92f:14034)
  • [Kol92] János Kollár, editor.
    Flips and abundance for algebraic threefolds.
    Société Mathématique de France, Paris, 1992.
    Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992). MR 1225842 (94f:14013)
  • [Kol96] János Kollár.
    Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics.
    Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
  • [KMM92] János Kollár, Yoichi Miyaoka, and Shigefumi Mori.
    Rationally connected varieties.
    J. Algebraic Geom., 1(3):429-448, 1992. MR 1158625 (93i:14014)
  • [Lan04] Adrian Langer.
    Semistable sheaves in positive characteristic.
    Ann. of Math. (2), 159(1):251-276, 2004. MR 2051393 (2005c:14021)
  • [Laz04] Robert Lazarsfeld.
    Positivity in algebraic geometry.
    Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
  • [Miy85] Yoichi Miyaoka.
    Deformation of a morphism along a foliation.
    In S. Bloch, editor, Algebraic Geometry, volume 46 of Proceedings of Symposia in pure Mathematics, pages 245-269, Providence, Rhode Island, 1985. American Mathematical Society. MR 0927960 (89e:14011)
  • [Mor79] Shigefumi Mori.
    Projective manifolds with ample tangent bundles.
    Ann. of Math. (2), 110(3):593-606, 1979. MR 0554387 (81j:14010)
  • [Ses82] C. S. Seshadri.
    Fibrés vectoriels sur les courbes algébriques, volume 96 of Astérisque.
    Société Mathématique de France, Paris, 1982.
    Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980. MR 0699278 (85b:14023)
  • [War71] F. Warner.
    Foundations of Differentiable Manifolds and Lie Groups.
    Scott, Foresman and Company, Glenview, Illinois and London, 1971. MR 0295244 (45:4312)

Additional Information

Stefan Kebekus
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Luis Solá Conde
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Matei Toma
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany and Mathematical Institute of the Romanian Academy, Bucharest

Received by editor(s): May 13, 2005
Received by editor(s) in revised form: September 3, 2005, and September 29, 2005
Published electronically: May 25, 2006
Additional Notes: The authors thank their hosting institution, Universität zu Köln. The first two authors were supported in full or in part by the Forschungsschwerpunkt “Globale Methoden in der komplexen Analysis” of the Deutsche Forschungsgemeinschaft. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.

American Mathematical Society