Inspired by the computation of the Kodaira dimension of symmetric powers of a complex projective variety of dimension by Arapura and Archava, we study their analytic and algebraic hyperbolicity properties. First, we show that some (or equivalently any) is rationally connected (resp. special) if and only if so is (except when the core of 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 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.
Partant du calcul de la dimension de Kodaira des produits symétriques d’une variété projective complexe de dimension par Arapura et Archava, nous étudions leurs propriétés analytiques et algébriques. Tout d’abord, nous montrons qu’un (ou de manière équivalente tout) est rationnellement connexe (resp. spécial) si et seulement si l’est (sauf lorsque le cœur de 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 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.
Accepted:
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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
Benoît Cadorel 1; Frédéric Campana 1; Erwan Rousseau 2
@article{JEP_2022__9__381_0, author = {Beno{\^\i}t Cadorel and Fr\'ed\'eric Campana and Erwan Rousseau}, title = {Hyperbolicity and specialness of symmetric~powers}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {381--430}, publisher = {\'Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.185}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.185/} }
TY - JOUR AU - Benoît Cadorel AU - Frédéric Campana AU - Erwan Rousseau TI - Hyperbolicity and specialness of symmetric powers JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 381 EP - 430 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.185/ DO - 10.5802/jep.185 LA - en ID - JEP_2022__9__381_0 ER -
%0 Journal Article %A Benoît Cadorel %A Frédéric Campana %A Erwan Rousseau %T Hyperbolicity and specialness of symmetric powers %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 381-430 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.185/ %R 10.5802/jep.185 %G en %F JEP_2022__9__381_0
Benoît Cadorel; Frédéric Campana; Erwan Rousseau. Hyperbolicity and specialness of symmetric powers. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 381-430. doi : 10.5802/jep.185. https://jep.centre-mersenne.org/articles/10.5802/jep.185/
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