[Fibrations rationnellement connexes maximales et courbes mobiles]
Un résultat bien connu de Miyaoka affirme qu’une varieté projective est uniréglée si son fibré tangent restreint à une courbe intersection complète générale n’est pas nef. De plus, en utilisant la filtration de Harder-Narasimhan, on peut montrer que le choix d’une telle courbe induit des feuilletages rationnellement connexes de la variété. Dans cette note nous montrons qu’une courbe mobile peut être trouvée telle que la fibration rationnellement connexe maximale soit associée à un terme de la filtration de Harder-Narasimhan correspondante du fibré tangent.
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.
Keywords: Uniruled variety, maximal rationally connected fibration, movable curve, Harder-Narasimhan filtration
Mot clés : varieté uniréglée, fibration rationnellement connexe maximale, courbe mobile, filtration de Harder-Narasimhan
@article{AIF_2009__59_6_2359_0,
author = {Sol\'a Conde, Luis E. and Toma, Matei},
title = {Maximal rationally connected fibrations and movable curves},
journal = {Annales de l'Institut Fourier},
pages = {2359--2369},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {59},
number = {6},
year = {2009},
doi = {10.5802/aif.2493},
mrnumber = {2640923},
zbl = {1245.14050},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2493/}
}
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 SP - 2359 EP - 2369 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2493/ DO - 10.5802/aif.2493 LA - en ID - AIF_2009__59_6_2359_0 ER -
%0 Journal Article %A Solá Conde, Luis E. %A Toma, Matei %T Maximal rationally connected fibrations and movable curves %J Annales de l'Institut Fourier %D 2009 %P 2359-2369 %V 59 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2493/ %R 10.5802/aif.2493 %G en %F AIF_2009__59_6_2359_0
Solá Conde, Luis E.; Toma, Matei. Maximal rationally connected fibrations and movable curves. Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2359-2369. doi : 10.5802/aif.2493. http://www.numdam.org/articles/10.5802/aif.2493/
[1] The pseudo-effectuve cone of a compact Kähler manifold and varieties of negative Kodaira dimension (preprint math.AG/0405285)
[2] Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4), Volume 25 (1992) no. 5, pp. 539-545 | Numdam | MR | Zbl
[3] Geometric stability of the cotangent bundle and the universal cover of a projective manifold (preprint arXiv:math/0405093)
[4] Restrictions of semistable bundles on projective varieties, Comment. Math. Helv., Volume 59 (1984) no. 4, pp. 635-650 | DOI | MR | Zbl
[5] Families of rationally connected varieties, J. Amer. Math. Soc., Volume 16 (2003) no. 1, pp. 57-67 | DOI | MR | Zbl
[6] The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997 | MR | Zbl
[7] Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) (ICTP Lect. Notes), Volume 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 335-393 | MR | Zbl
[8] Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom., Volume 16 (2007) no. 1, pp. 65-81 | DOI | MR | Zbl
[9] Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 32, Springer-Verlag, Berlin, 1996 | MR | Zbl
[10] Rationally connected varieties, J. Algebraic Geom., Volume 1 (1992) no. 3, pp. 429-448 | MR | Zbl
[11] Semistable sheaves on projective varieties and their restriction to curves, Math. Ann., Volume 258 (1981/82) no. 3, pp. 213-224 | DOI | MR | Zbl
[12] Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proc. Sympos. Pure Math.), Volume 46, Amer. Math. Soc., Providence, RI, 1987, pp. 245-268 | MR | Zbl
[13] A numerical criterion for uniruledness, Ann. of Math. (2), Volume 124 (1986) no. 1, pp. 65-69 | DOI | MR | Zbl
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