Related papers: Mesures et \'equidistribution sur les espaces de B…

We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution…

In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabr\'e's proof of the classical isoperimetric inequality in…

Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer $g \geq 2$, we…

For a projective variety X defined over a field K, there is a special class of self-morphisms of X called algebraic dynamical systems. In this paper we take K to be the function field of a smooth curve and prove that at each place of K,…

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For…

In the paper, we generalize the Arzel\`a-Ascoli theorem in the setting of uniform spaces. At first, we recall well-known facts and theorems coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of…

Differentiability of geometric and arithmetic volumes of Hermitian line-bundles leads to the proof of equidistribution results on projective varieties using the variational principle. In this article, we work in the setting of adelic…

We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system,…

We prove equidistribution of Weierstrass points on Berkovich curves. Let $X$ be a smooth proper curve of positive genus over a complete algebraically closed non-Archimedean field $K$ of equal characteristic zero with a non-trivial…

In this paper, we prove a uniform version of Poonen's "Mordell-Lang Plus Bogomolov" theorem for abelian varieties. We mainly generalize R\'emond's work on large points to allow an extra $\epsilon$-neighborhood. The part on small points…

We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space over a complete non-archimedean field. As an application, we give a sufficient condition for equidistribution in terms of a…

The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4…

Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich…

We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A…

We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T.…

In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in…

In this article, we prove the boundedness of minimal slopes of adelic line bundles over function fields of characteristic 0. This can be applied to prove the equidistribution of generic and small points with respect to a big and…

Recently, R\'emond stated a very general conjecture on lower bounds of a normalized height on either an abelian variety or a power of the multiplicative group. In this note, we extend a particular case of this conjecture to split…

We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those…

Although Berkovich spaces may fail to be metrizable when defined over too big a field, we prove that a large part of their topology can be recovered through sequences: for instance, limit points of subsets are actual limits of sequences and…