Twisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliations

[Faisceaux cotangents tordus et un théorème de Kobayashi-Ochiai pour les feuilletages]

Höring, Andreas

Annales de l'Institut Fourier, Tome 64 (2014), p. 2465-2480 / Harvested from Numdam

Résumé

Soit $X$ une variété projective normale et $A$ un diviseur de Cartier ample sur $X$. Supposons que $X$ n’est pas l’espace projectif. Nous montrons que le faisceau cotangent tordu $\Omega _X \otimes A$ est génériquement nef par rapport à la polarisation $A$. Comme conséquence nous obtenons un théorème de Kobayashi-Ochiai pour les feuilletages  : si $\mathcal{F} \subsetneq T_X$ est un feuilletage tel que $\det \mathcal{F} \equiv i_{\mathcal{F}} A$, alors $i_{\mathcal{F}}$ est au plus le rang de $\mathcal{F}$.

Abstract

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$. Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf $\Omega _X \otimes A$ is generically nef with respect to the polarisation $A$. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $\mathcal{F} \subsetneq T_X$ is a foliation such that $\det \mathcal{F} \equiv i_{\mathcal{F}} A$, then $i_{\mathcal{F}}$ is at most the rank of $\mathcal{F}$.

Détail

Publié le : 2014-01-01

MR 3331171 | Zbl 06387344

DOI : https://doi.org/10.5802/aif.2917
Classification:  14F10,  37F75,  14M22,  14E30,  14J40
Mots clés: faisceau cotangent, feuilletages, théorème de Kobayashi-Ochiai

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