Maximal rationally connected fibrations and movable curves
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, p. 2359-2369

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.

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.

DOI : https://doi.org/10.5802/aif.2493
Classification:  14J99,  14D99,  53C12
Keywords: Uniruled variety, maximal rationally connected fibration, movable curve, Harder-Narasimhan filtration
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     author = {Sol\'a Conde, Luis E. and Toma, Matei},
     title = {Maximal rationally connected fibrations and movable curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     pages = {2359-2369},
     doi = {10.5802/aif.2493},
     zbl = {1245.14050},
     mrnumber = {2640923},
     zbl = {pre05673899},
     language = {en},
     url = {http://www.numdam.org/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, Volume 59 (2009) no. 6, pp. 2359-2369. doi : 10.5802/aif.2493. http://www.numdam.org/item/AIF_2009__59_6_2359_0/

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