Carlo Gasbarri | Université de Strasbourg (original) (raw)

Papers by Carlo Gasbarri

International Mathematics Research Notices, Dec 17, 2020

Let p be an algebraic point of a projective variety X defined over a number field. Liouville ineq... more Let p be an algebraic point of a projective variety X defined over a number field. Liouville inequality tells us that the norm at p of a non vanishing integral global section of an hermitian line bundle over X is either zero or it cannot be too small with respect to the sup norm of the section itself. We study inequalities similar to Liouville's for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

Cette these est composee de trois chapitres, traitants des questions liees a la geometrie d'a... more Cette these est composee de trois chapitres, traitants des questions liees a la geometrie d'arakelov des varietes arithmetiques ; plus en particulier de questions liees aux choix des metriques sur les fibres sur telles varietes. Dans le premier chapitre on prouve un analogue, dans le cadre de la theorie d'arakelov, du theoreme d'annulation de serre en cohomologie ; il s'agit d'un theoreme d'annulation asymptotique du dernier groupe de cohomologie d'un fibre sur une variete arithmetique projective ayant la fibre generique de cohen-macaulay. Dans le deuxieme chapitre on prouve que la seule theorie de l'intersection sur une surface arithmetique qui etend l'accouplement de neron-tate sur les diviseurs de degre zero est la theorie d'arakelov originale, a savoir celle obtenue n'utilisant que des metriques permises a l'infini. Dans le troisieme chapitre on construit une hauteur sur l'espace de modules des fibres stables de rang et determi...

American Journal of Mathematics, 2005

This paper addresses conjectures of E. Bombieri and P. Vojta in the special case of ruled surface... more This paper addresses conjectures of E. Bombieri and P. Vojta in the special case of ruled surfaces not birational to P 2. Apart from this implicit restriction to P 1 bundles S over an elliptic curve, the ultimate question of the arithmetic of pairs (S, D) for a divisor D requires further restrictions on D which turn the proposed conjectures into the study of Roth's theorem on approximation of algebraic numbers α, but for α now parametrized by an elliptic curve. With these restrictions, best possible answers are obtained. The same study may also be carried out for holomorphic maps, and this is done simultaneously.

Contemporary Mathematics, 2015

This mini-course described the Thue-Siegel method, as used in the proof of Faltings' theorem on t... more This mini-course described the Thue-Siegel method, as used in the proof of Faltings' theorem on the Mordell conjecture. The exposition followed Bombieri's variant of this proof, which avoids the machinery of Arakelov theory.

Dans ce cours, nous nous proposons d’expliquer comment des theoremes d’algebrisation classiques, ... more Dans ce cours, nous nous proposons d’expliquer comment des theoremes d’algebrisation classiques, concernant des varietes ou des faisceux coherents analytiques, possedent des avatars en geometrie formelle et en geometrie diophantienne. Nous mettrons l’accent sur les points communs entre les preuves de ces differents theoremes, et sur leurs consequences "concretes" concernant la geometrie et l’arithmetique des varietes algebriques. 1. Algebrisation de sous-schemas formels de varietes projectives. 2. Theoremes de Lefschetz et geometrie formelle: les theoremes de Grauert et de Grothendieck. 3. Algebrisation en geometrie diophantienne. 4. Applications aux feuilletages.

We will explain some results about the arithmetic structure of algebraic points over a variety de... more 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.

Varietes presque rationnelles, leurs points rationnels et leurs degenerescences.- Topics in Dioph... more Varietes presque rationnelles, leurs points rationnels et leurs degenerescences.- Topics in Diophantine Equations.- Diophantine Approximation and Nevanlinna Theory.

HAL (Le Centre pour la Communication Scientifique Directe), 2021

After reviewing the proof by Bogomolov of the fact that, on smooth projective surfaces with c 2 1... more After reviewing the proof by Bogomolov of the fact that, on smooth projective surfaces with c 2 1 > c 2 , curves of bounded geometric genus form a bounded family, we explain the main steps of the proof, given by McQuillan, of the Green-Griffiths conjectures for these surfaces. Observed from far, the two proofs follow the same strategy but the second requires a much deeper analysis of the tools involved. In order to describe the McQuillan proof, we explain the construction of the Ahlfors currents associated to entire curves in variety and we show how these can be used to produce a substitute of intersection numbers. A proof of the tautological inequality either in the standard then in the logarithmic case is given. We will explain how the hypothesis imply that we may suppose that the involved entire curve (which is supposed to not exist) is the leaf of a foliation. In order to simplify some technical points of the proof, we will make some restrictions on the singularities of this foliation (the general case requires a deeper technical analysis but the main ideas of the proof are already visible under this restriction). In the last section we give a very brief description of a possible strategy (proposed by McQuillan) for the proof of the general case of the conjecture; we will also explain the main difficulties one would need to overcome.

In these notes we give a reasonably self contained proof of three of the main theorems of the dio... more In these notes we give a reasonably self contained proof of three of the main theorems of the diophantine geometry of curves over function fields of characteristic zero. Let F be a function field of dimension one over the field of the complex numbers C i. e. a field of transcendence degree one over C. Let X F be a smooth projective curve over F. We prove that:-if the genus of X F is zero then it is isomorphic, over F , to the projective line P 1 .-If the genus of X F is one and X F is not isomorphic (over the algebraic closure of F) to a curve defined over C, then the set of F-rational points of X F has the natural structure of a finitely generated abelian group (Theorem of Mordell Weil).-If the genus of X F is strictly bigger then one and X F is not isomorphic (over the algebraic closure of F) to a curve defined over C, then the set of F-rational points of X F is finite (former Mordell conjecture). The proofs use only standard algebraic geometry, basic topology and analysis of algebraic surfaces (all the background can be found in standard texts as [11] or [8]).

arXiv: Algebraic Geometry, 2016

The analogy between the arithmetic of varieties over number fields and the arithmetic of varietie... more The analogy between the arithmetic of varieties over number fields and the arithmetic of varieties over function fields is a leading theme in arithmetic geometry. This analogy is very powerful but there are some gaps. In this note we will show how the presence of isotrivial varieties over function fields (the analogous of which do not seems to exist over number fields) breaks this analogy. Some counterexamples to a statement similar to Northcott Theorem are proposed. In positive characteristic, some explicit counterexamples to statements similar to Lang and Vojta conjectures are given.

We give a generalization of the classical Bombieri--Schneider--Lang criterion in transcendence th... more We give a generalization of the classical Bombieri--Schneider--Lang criterion in transcendence theory. We give a local notion of LG--germ, which is similar to the notion of E-- function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let K⊂ C be a number field and X a quasi--projective variety defined over K. Let γ M→ X be an holomorphic map of finite order from a parabolic Riemann surface to X such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every p∈ X(K)∩γ(M) the formal germ of M near P is an LG-- germ, then we prove that X(K)∩γ(M) is a finite set. Then we define the notion of conformally parabolic Khäler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact Khäler manifold is conformally parabolic; in particular every quasi projective var...

Let p be an algebraic point of a projective variety X defined over a number field. Liouville ineq... more Let p be an algebraic point of a projective variety X defined over a number field. Liouville inequality tells us that the norm at p of a non vanishing integral global section of an hermitian line bundle over X is either zero or it cannot be too small with respect to the norm of the section itself. We study inequalities similar to Liouville's for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnowsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective m... more We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective manifolds. Using Ahlfors currents in higher dimension, we obtain several strong degeneracy statements.

Liouville inequality is a lower bound of the norm of an integral section of a line bundle on an a... more 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

manuscripta mathematica, 2021

One of the main objectives in mathematics is solving algebraic equations. Given a system of algeb... more One of the main objectives in mathematics is solving algebraic equations. Given a system of algebraic equations G(X) = 0 defined over a ring A, may we know if there are solutions of it on A? and, in case, may we explicitly find them? Of course the prototype of such a ring is Z or rings finite over it, but the theorem of Matiyasevich tells us that we cannot find a general method to answer to the question in this situation. Never the less one can try to see if it is possible to answer to the question for some class of systems of algebraic equations. Since long time we know that there is an interesting analogy between the ring Z and the ring k[t] where k is a field: – They are both principal ideal domains with Krull dimension one. – In both rings a product formula holds: Over Z the product of the all the possible absolute values of an element is one; over k[t] every rational functions has as many poles as many zeroes on the projective line. Essentially all the technics one can develop ...

Duke Mathematical Journal, 2020

International Mathematics Research Notices, Dec 17, 2020

American Journal of Mathematics, 2005

Contemporary Mathematics, 2015

HAL (Le Centre pour la Communication Scientifique Directe), 2021

arXiv: Algebraic Geometry, 2016

Let p be an algebraic point of a projective variety X defined over a number field. Liouville ineq... more Let p be an algebraic point of a projective variety X defined over a number field. Liouville inequality tells us that the norm at p of a non vanishing integral global section of an hermitian line bundle over X is either zero or it cannot be too small with respect to the norm of the section itself. We study inequalities similar to Liouville's for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnowsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

manuscripta mathematica, 2021

Duke Mathematical Journal, 2020