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.
@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},
volume = {59},
year = {2009},
pages = {2359-2369},
doi = {10.5802/aif.2493},
zbl = {pre05673899},
mrnumber = {2640923},
zbl = {1245.14050},
language = {en},
url = {http://dml.mathdoc.fr/item/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) pp. 2359-2369. doi : 10.5802/aif.2493. http://gdmltest.u-ga.fr/item/AIF_2009__59_6_2359_0/
[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), Tome 25 (1992) no. 5, pp. 539-545 | Numdam | MR 1191735 | Zbl 0783.14022
[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., Tome 59 (1984) no. 4, pp. 635-650 | Article | MR 780080 | Zbl 0599.14015
[5] Families of rationally connected varieties, J. Amer. Math. Soc., Tome 16 (2003) no. 1, pp. 57-67 | Article | MR 1937199 | Zbl 1092.14063
[6] The geometry of moduli spaces of sheaves, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics, E31 (1997) | MR 1450870 | Zbl 0872.14002
[7] Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), Abdus Salam Int. Cent. Theoret. Phys., Trieste (ICTP Lect. Notes) Tome 6 (2001), pp. 335-393 | MR 1919462 | Zbl 1086.14506
[8] Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom., Tome 16 (2007) no. 1, pp. 65-81 | Article | MR 2257320 | Zbl 1120.14011
[9] Rational curves on algebraic varieties, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Tome 32 (1996) | MR 1440180 | Zbl 0877.14012
[10] Rationally connected varieties, J. Algebraic Geom., Tome 1 (1992) no. 3, pp. 429-448 | MR 1158625 | Zbl 0780.14026
[11] Semistable sheaves on projective varieties and their restriction to curves, Math. Ann., Tome 258 (1981/82) no. 3, pp. 213-224 | Article | MR 649194 | Zbl 0473.14001
[12] Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 46 (1987), pp. 245-268 | MR 927960 | Zbl 0659.14008
[13] A numerical criterion for uniruledness, Ann. of Math. (2), Tome 124 (1986) no. 1, pp. 65-69 | Article | MR 847952 | Zbl 0606.14030