Maximal rationally connected fibrations and movable curves

Luis E. Solá Conde[1]; Matei Toma[2]

  • [1] Universidad Rey Juan Carlos Departamento de Matemáticas C. Tulipán s.n. 28933 Móstoles, Madrid (Spain)
  • [2] Institut de Mathématiques Elie Cartan Nancy-Université B.P. 239 54506 Vandoeuvre-lès-Nancy Cedex (France) and Institute of Mathematics of the Romanian Academy

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 6, page 2359-2369
  • ISSN: 0373-0956

Abstract

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A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.

How to cite

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Solá Conde, Luis E., and Toma, Matei. "Maximal rationally connected fibrations and movable curves." Annales de l’institut Fourier 59.6 (2009): 2359-2369. <http://eudml.org/doc/10457>.

@article{SoláConde2009,
abstract = {A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.},
affiliation = {Universidad Rey Juan Carlos Departamento de Matemáticas C. Tulipán s.n. 28933 Móstoles, Madrid (Spain); Institut de Mathématiques Elie Cartan Nancy-Université B.P. 239 54506 Vandoeuvre-lès-Nancy Cedex (France) and Institute of Mathematics of the Romanian Academy},
author = {Solá Conde, Luis E., Toma, Matei},
journal = {Annales de l’institut Fourier},
keywords = {Uniruled variety; maximal rationally connected fibration; movable curve; Harder-Narasimhan filtration; uniruled variety; rationally connected fibration},
language = {eng},
number = {6},
pages = {2359-2369},
publisher = {Association des Annales de l’institut Fourier},
title = {Maximal rationally connected fibrations and movable curves},
url = {http://eudml.org/doc/10457},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Solá Conde, Luis E.
AU - Toma, Matei
TI - Maximal rationally connected fibrations and movable curves
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2359
EP - 2369
AB - A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.
LA - eng
KW - Uniruled variety; maximal rationally connected fibration; movable curve; Harder-Narasimhan filtration; uniruled variety; rationally connected fibration
UR - http://eudml.org/doc/10457
ER -

References

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  1. Sebastien Boucksom, Jean-Pierre Demailly, Mihai Păun, Thomas Peternell, The pseudo-effectuve cone of a compact Kähler manifold and varieties of negative Kodaira dimension Zbl1267.32017
  2. Frédéric Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), 539-545 Zbl0783.14022MR1191735
  3. Frédéric Campana, Thomas Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold Zbl1218.14030
  4. Hubert Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59 (1984), 635-650 Zbl0599.14015MR780080
  5. Tom Graber, Joe Harris, Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 57-67 Zbl1092.14063MR1937199
  6. Daniel Huybrechts, Manfred Lehn, The geometry of moduli spaces of sheaves, (1997), Friedr. Vieweg & Sohn, Braunschweig Zbl0872.14002MR1450870
  7. Jun-Muk Hwang, Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) 6 (2001), 335-393, Abdus Salam Int. Cent. Theoret. Phys., Trieste Zbl1086.14506MR1919462
  8. Stefan Kebekus, Luis Solá Conde, Matei Toma, Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom. 16 (2007), 65-81 Zbl1120.14011MR2257320
  9. János Kollár, Rational curves on algebraic varieties, 32 (1996), Springer-Verlag, Berlin Zbl0877.14012MR1440180
  10. János Kollár, Yoichi Miyaoka, Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), 429-448 Zbl0780.14026MR1158625
  11. Vikram B. Mehta, Annamalai Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1981/82), 213-224 Zbl0473.14001MR649194
  12. Yoichi Miyaoka, Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) 46 (1987), 245-268, Amer. Math. Soc., Providence, RI Zbl0659.14008MR927960
  13. Yoichi Miyaoka, Shigefumi Mori, A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), 65-69 Zbl0606.14030MR847952

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