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# Documents  Gasbarri, Carlo | enregistrements trouvés : 12

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## Improved bounds for jet differential operators and the Kobayashi conjecture Demailly, Jean-Pierre | CIRM H

Post-edited

Research talks;Algebraic and Complex Geometry

We will discuss several new ideas that can show the existence of jet differential operators on arbitrary projective varieties, and also on general hypersurfaces of $\mathbb{P}^n$ of sufficiently high degree. These results can be applied to improve degree bounds in several hyperbolicity problems and especially in the proof of the Kobayashi conjecture.

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## Arithmetic geometry.Lectures given at the C.I.M.E. summer schoolCetraro # september 10-15, 2007 Corvaja, Pietro ; Gasbarri, Carlo ; Colliot-Thélène, Jean-Louis ; Swinnerton-Dyer , Peter ; Vojta, Paul | Springer;Fondazione CIME 2011

Congrès

- xi; 232 p.
ISBN 978-3-642-15944-2

Lecture notes in mathematics , 2009

Localisation : Collection 1er étage

géométrie algébrique arithmétique # analyse diophantienne

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## Rational points, rational curves, and entire holomorphic curves on projective varieties.CRM short thematic program on rational points, rational curves, and entire holomorphic curves and algebraic varietiesMontréal # June 3-28, 2013 Gasbarri, Carlo ; Lu, Steven ; Roth, Mike ; Tschinkel, Yuri | American Mathematical Society;Centre De Recherches Mathematiques 2015

Congrès

- vii; 165 p.
ISBN 978-1-4704-1458-0

Contemporary mathematics , 0654

Localisation : Collection 1er étage

géométrie algébrique # point rationnel # courbe rationnelle # courbe holomorphe entière # variété algébrique # approximation Diophantienne

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## Liouville's inequality for transcendental points on projective varieties Gasbarri, Carlo | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry;Number Theory

Liouville inequality is a lower bound of the norm of an integral section of a line bundle on an algebraic point of a variety. It is an important tool in may proofs in diophantine geometry and in transcendence. On transcendental points an inequality as good as Liouville inequality cannot hold. We will describe similar inequalities which hold for "many" transcendental points and some applications

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## A Grassmannian technique and the Kobayashi Conjecture Riedl, Eric | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

An entire curve on a complex variety is a holomorphic map from the complex numbers to the variety. We discuss two well-known conjectures on entire curves on very general high-degree hypersurfaces $X$ in $\mathbb{P}^n$: the Green-Griffiths-Lang Conjecture, which says that the entire curves lie in a proper subvariety of $X$, and the Kobayashi Conjecture, which says that X contains no entire curves.
We prove that (a slightly strengthened version of) the Green-Griffiths-Lang Conjecture in dimension $2n$ implies the Kobayashi Conjecture in dimension $n$. The technique has already led to improved bounds for the Kobayashi Conjecture. This is joint work with David Yang.
An entire curve on a complex variety is a holomorphic map from the complex numbers to the variety. We discuss two well-known conjectures on entire curves on very general high-degree hypersurfaces $X$ in $\mathbb{P}^n$: the Green-Griffiths-Lang Conjecture, which says that the entire curves lie in a proper subvariety of $X$, and the Kobayashi Conjecture, which says that X contains no entire curves.
We prove that (a slightly strengthened version ...

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## Arithmetic and algebraic hyperbolicity Javanpeykar, Ariyan | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

The Green-Griffiths-Lang-Vojta conjectures relate the hyperbolicity of an algebraic variety to the finiteness of sets of “rational points”. For instance, it suggests a striking answer to the fundamental question “Why do some polynomial equations with integer coefficients have only finitely many solutions in the integers?”. Namely, if the zeroes of such a system define a hyperbolic variety, then this system should have only finitely many integer solutions.
In this talk I will explain how to verify some of the algebraic, analytic, and arithmetic predictions this conjecture makes. I will present results that are joint work with Ljudmila Kamenova.
The Green-Griffiths-Lang-Vojta conjectures relate the hyperbolicity of an algebraic variety to the finiteness of sets of “rational points”. For instance, it suggests a striking answer to the fundamental question “Why do some polynomial equations with integer coefficients have only finitely many solutions in the integers?”. Namely, if the zeroes of such a system define a hyperbolic variety, then this system should have only finitely many integer ...

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## ​Special manifolds, the core fibration, rational and entire curves Campana, Frédéric | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

For complex projective manifolds $X$ of general type, Lang claimed the equivalence between three fields: birational geometry, complex hyperbolicity, and arithmetic. We extend this equivalence to arbitrary $X$’s by introducing the (antithetical) class of “Special” manifolds and constructing the “Core” fibration, the unique one with special fibres and general type “orbifold” base. We conjecture that special manifolds —which are defined algebro-geometrically by a certain non-positivity of their cotangent bundles— are also exactly the ones having Zariski-dense entire curves (so violating the GGL property). We shall give (j.w. J. Winkelmann) some examples supporting this conjecture. The arithmetic aspect will be skipped. For complex projective manifolds $X$ of general type, Lang claimed the equivalence between three fields: birational geometry, complex hyperbolicity, and arithmetic. We extend this equivalence to arbitrary $X$’s by introducing the (antithetical) class of “Special” manifolds and constructing the “Core” fibration, the unique one with special fibres and general type “orbifold” base. We conjecture that special manifolds —which are defined al...

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## ​​​Isotriviality for families given by regular foliations Amerik, Ekaterina | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

Viehweg and Zuo obtained several results concerning the moduli number in smooth families of polarized varieties with semi-ample canonical class over a quasiprojective base. These results led Viehweg to conjecture that the base of a family of maximal variation is of log-general type, and the conjecture has been recently proved by Campana and Paun.
From the “opposite” side, Taji proved that a smooth projective family over a special (in the sense of Campana) quasiprojective base is isotrivial.
We extend Taji’s theorem to quasismooth families, that is, families of leaves of compact foliations without singularities. This is a joint work with F. Campana
Viehweg and Zuo obtained several results concerning the moduli number in smooth families of polarized varieties with semi-ample canonical class over a quasiprojective base. These results led Viehweg to conjecture that the base of a family of maximal variation is of log-general type, and the conjecture has been recently proved by Campana and Paun.
From the “opposite” side, Taji proved that a smooth projective family over a special (in the sense ...

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## Arithmetic of algebraic points on varieties over function fields - Part 1 Gasbarri, Carlo | CIRM H

Single angle

Research talks;Algebraic and Complex Geometry

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields. We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some ...

14G40

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## Arithmetic of algebraic points on varieties over function fields - Part 2 Gasbarri, Carlo | CIRM H

Single angle

Research talks;Algebraic and Complex Geometry

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields. We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some ...

14G40

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## Arithmetic of algebraic points on varieties over function fields - Part 3 Gasbarri, Carlo | CIRM H

Single angle

Research talks;Algebraic and Complex Geometry

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields. We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some ...

14G40

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## Arithmetic of algebraic points on varieties over function fields - Part 4 Gasbarri, Carlo | CIRM H

Single angle

Research talks;Algebraic and Complex Geometry

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields. We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some ...

14G40

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