Maximal rationally connected fibrations and movable curves. (English) Zbl 1245.14050

A proper algebraic variety \(X\) is called uniruled (resp., rationally connected) if there exists a rational curve through a general point \(x\in X\) (resp., a general pair of points \(x_1,x_2\in X\)). By the work of J. Kollár, Y. Miyaoka and S. Mori [J. Algebr. Geom. 1, No. 3, 429–448 (1992; Zbl 0780.14026)] and F. Campana [Ann. Sci. Éc. Norm. Supér. (4) 25, No. 5, 539–545 (1992; Zbl 0783.14022)], it is known that any uniruled variety \(X\) admits a canonical maximal rationally connected (MRC) fibration \(X\dashrightarrow Z\). Such an MRC fibration gives a foliation \(G\subset T_X\). The paper under review proves that there exists a movable curve \(C\) in \(X\) such that \(G\) appears as a step of the Harder-Narasimhan filtrations of \(T_X\) with respect to the curve class of \(C\).


14M22 Rationally connected varieties
14J99 Surfaces and higher-dimensional varieties
53C12 Foliations (differential geometric aspects)
Full Text: DOI arXiv EuDML


[1] Boucksom, Sebastien; Demailly, Jean-Pierre; Păun, Mihai; Peternell, Thomas, The pseudo-effectuve cone of a compact Kähler manifold and varieties of negative Kodaira dimension · Zbl 1267.32017
[2] Campana, Frédéric, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4), 25, 5, 539-545 (1992) · Zbl 0783.14022
[3] Campana, Frédéric; Peternell, Thomas, Geometric stability of the cotangent bundle and the universal cover of a projective manifold · Zbl 1218.14030
[4] Flenner, Hubert, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv., 59, 4, 635-650 (1984) · Zbl 0599.14015 · doi:10.1007/BF02566370
[5] Graber, Tom; Harris, Joe; Starr, Jason, Families of rationally connected varieties, J. Amer. Math. Soc., 16, 1, 57-67 (2003) · Zbl 1092.14063 · doi:10.1090/S0894-0347-02-00402-2
[6] Huybrechts, Daniel; Lehn, Manfred, The geometry of moduli spaces of sheaves (1997) · Zbl 0872.14002
[7] Hwang, Jun-Muk, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 6, 335-393 (2001) · Zbl 1086.14506
[8] Kebekus, Stefan; Solá Conde, Luis; Toma, Matei, Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom., 16, 1, 65-81 (2007) · Zbl 1120.14011 · doi:10.1090/S1056-3911-06-00435-8
[9] Kollár, János, Rational curves on algebraic varieties, 32 (1996) · Zbl 0877.14012
[10] Kollár, János; Miyaoka, Yoichi; Mori, Shigefumi, Rationally connected varieties, J. Algebraic Geom., 1, 3, 429-448 (1992) · Zbl 0780.14026
[11] Mehta, Vikram B.; Ramanathan, Annamalai, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann., 258, 3, 213-224 (198182) · Zbl 0473.14001 · doi:10.1007/BF01450677
[12] Miyaoka, Yoichi, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 46, 245-268 (1987) · Zbl 0659.14008
[13] Miyaoka, Yoichi; Mori, Shigefumi, A numerical criterion for uniruledness, Ann. of Math. (2), 124, 1, 65-69 (1986) · Zbl 0606.14030 · doi:10.2307/1971387
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.