Rencontre en hyperbolicité

Date/heure
5 octobre 2021 - 6 octobre 2021
Toute la journée

Lieu
Salle de conférences Nancy

Catégorie d'évènement
Conférence


Le 5 et 6 octobre 2021, l’IECL accueille est un mini workshop autour de l’hyperbolicité en géométrie complexe.

 

Mardi 5 :
  • 10h00-10h30 : Accueil et café
  • 10h30-11h30 : Antoine Étesse :
    Application des wronskiens géométriques généralisés en hyperbolicité.
  • 11h30-15h00 : Déjeuner
  • 15h00-16h00 : Ya Deng
    Picard hyperbolicity of manifolds admitting harmonic bundles, I
  • 16h00-16h30 : Pause café
  • 16h30-17h30 : Benoît Cadorel
    Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles, II  (j.w. Y. Deng)
Mercredi 6 :
  • 9h00-9h30 : Pause café
  • 9h30-10h30 : Erwan Rousseau :
    Numerically special varieties.
  • 10h30-11h00 : Pause café
  • 11h00-12h00 : Jörg Winkelmann
  • 12h00-14h00 : Déjeuner
  • 14h00-14h30 : Pause café

 

Résumés des exposés :

 

Benoît Cadorel : Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles, II  (j.w. Y. Deng)

We will explain how to use the criterion introduced by Ya Deng in his talk, to obtain pseudo-hyperbolicity results for open varieties U supporting nilpotent harmonic bundles.  As we will see, this generalizes earlier work of Y. Brunebarbe, D. Brotbek, Y. Deng… concerning quotients of bounded symmetric domains, or more generally, varieties supporting a variation of Hodge structures.

As in the case of bounded symmetric domains or variations of integral polarized Hodge structures, it is possible to pass to an étale cover U’ over U, to obtain the pseudo-hyperbolicity of any compactification X’ of U’. However, the situation is a bit more complicated in our setting, since it does not seem possible in general to obtain arbitrarily high ramification orders around every component of the boundary D’ of X’.  We will show how to overcome this difficulty, by remarking that there is actually a dichotomy between the components of D’ with high multiplicity, and other ones over which the harmonic bundle extends.

 

Ya Deng :  Picard hyperbolicity of manifolds admitting harmonic bundles, I

 This is the first part of a joint work with Benoit Cadorel. Picard hyperbolicity is a notion which generalizes the Big Picard theorem. It fascinates me since it implies the algebraicity of analytic maps, first studied by Ariyan Javanpeykar and Kucharczyk. There are many interesting progress on this topic by many specialists, including several speakers in the workshop. In this talk I will explain some new criterion of Picard hyperbolicity for quasi-compact Kahler manifolds whose holomorphic sectional curvature is negative. Such criterion will be used to show the Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles, which will be explained in the talk of Benoit. The proof is Nevanlinna theoretic, and there are two main ingredients: one is the Nevanlinna characteristic function for positive currents, introduced in the work of Brunebarbe-Brotbek; the other one is the use of Skoda extension of pseudo Kahler forms over the Kahler compactification. If time allows, I will briefly explain some criterion for tautness, whose proof is in a similar vein.

 

Antoine Étesse : Application des wronskiens géométriques généralisés en hyperbolicité.

Lors de cet exposé, nous rappellerons la définition des Wronskiens généralisés, et en exhiberons une sous-famille, dont les éléments sont dits géométriques. Ces Wronskiens généralisés géométriques ont deux avantages: d’une part, ils permettent des constructions géométriques globales, que l’on précisera, et d’autre part, ils permettent toujours de caractériser l’indépendance linéaire des fonctions holomorphes (ce qui constitue la propriété fondamentale des Wronskiens généralisés, connue depuis Roth au moins (∼1950)). Nous présenterons alors une applications de cette construction en hyperbolicité, et plus précisément à l’étude des familles de courbes entières sur des hypersurfaces de Fermat.

 

Erwan Rousseau : Numerically special varieties.

Campana introduced the class of special varieties as the varieties admitting no maps onto an orbifold of general type. They are also characterized by the non-existence of Bogomolov sheaves which are rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana has conjectured that one can replace the Kodaira dimension by the numerical dimension in this characterization. We prove partially this conjecture showing that a projective manifold admitting a rank one coherent subsheaf of the cotangent bundle with numerical dimension 1 is not special. This is a joint work with J.V. Pereira and F. Touzet.