Hyperbolicity and specialness of symmetric powers

Benoît Cadorel; Frédéric Campana; Erwan Rousseau

doi:10.5802/jep.185

Hyperbolicity and specialness of symmetric powers
[Produits symétriques de type hyperbolique et spécial]

Benoît Cadorel ¹ ; Frédéric Campana ¹ ; Erwan Rousseau ²

¹ Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, Site de Nancy B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France
² Institut Universitaire de France & Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique 6, avenue Victor Le Gorgeu, 29238 Brest Cedex 3, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 381-430.

Résumé
Abstract

Partant du calcul de la dimension de Kodaira des produits symétriques $X_{m}$ d’une variété projective complexe $X$ de dimension $n \geq 2$ par Arapura et Archava, nous étudions leurs propriétés analytiques et algébriques. Tout d’abord, nous montrons qu’un (ou de manière équivalente tout) $X_{m}$ est rationnellement connexe (resp. spécial) si et seulement si $X$ l’est (sauf lorsque le cœur de $X$ est une courbe dans le cas spécial). Ensuite nous construisons des courbes entières denses dans les produits (suffisamment grands) de surfaces K3 et des produits de courbes. Nous donnons également un critère fondé sur la positivité des fibrés de différentielles de jets qui implique la pseudo-hyperbolicité des produits symétriques. Comme application, nous établissons l’hyperbolicité au sens de Kobayashi des produits symétriques des hypersurfaces projectives génériques de degré suffisamment grand. Du côté algébrique, nous donnons un critère impliquant que les sous-variétés de codimension $\leq n - 2$ des produits symétriques sont de type général. Il s’applique en particulier aux variétés à cotangent ample. Finalement, nous utilisons une approche métrique pour l’étude des produits symétriques des quotients de la boule.

Inspired by the computation of the Kodaira dimension of symmetric powers $X_{m}$ of a complex projective variety $X$ of dimension $n \geq 2$ by Arapura and Archava, we study their analytic and algebraic hyperbolicity properties. First, we show that some (or equivalently any) $X_{m}$ is rationally connected (resp. special) if and only if so is $X$ (except when the core of $X$ is a curve in the case of specialness). Then we construct dense entire curves in (sufficiently high) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersurfaces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension $\leq n - 2$ of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, we use a metric approach to study symmetric powers of ball quotients.

Reçu le : 2021-07-16
Accepté le : 2022-02-05
Publié le : 2022-02-16

DOI : 10.5802/jep.185

Classification : 14J15, 32Q45
Keywords: Green-Griffiths-Lang conjectures, complex hyperbolic varieties, special varieties, symmetric products
Mot clés : Conjectures de Green-Griffiths-Lang, variétés hyperboliques complexes, variétés spéciales, produits symétriques

Affiliations des auteurs :

Benoît Cadorel ¹ ; Frédéric Campana ¹ ; Erwan Rousseau ²

¹ Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, Site de Nancy B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France
² Institut Universitaire de France & Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique 6, avenue Victor Le Gorgeu, 29238 Brest Cedex 3, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Hyperbolicity and specialness of symmetric~powers},
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Benoît Cadorel; Frédéric Campana; Erwan Rousseau. Hyperbolicity and specialness of symmetric powers. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 381-430. doi : 10.5802/jep.185. https://jep.centre-mersenne.org/articles/10.5802/jep.185/

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