arXiv subjects

François Ballaÿ

Publications and source records attributed to François Ballaÿ.

4 recordsLinked to original sources

arXiv2025

Approximation of adelic divisors and equidistribution of small points

We study the asymptotic distribution of the Galois orbits of generic sequences of algebraic points of small height in a projective variety over a number field. Our main result is a generalization of Yuan's equidistribution theorem that applies to heights for which Zhang's lower bound for the essential minimum is not necessarily an equality. It extends to all projective varieties a theorem of Burgos Gil, Philippon, Rivera-Letelier and the second author for toric varieties. It also applies to sums of canonical heights for an algebraic dynamical system, and in particular it recovers Kühne's semiabelian equidistribution theorem. We also generalize previous work of Chambert-Loir and Thuillier to obtain new logarithmic equidistribution results. Finally we extend our main result to the quasi-projective setting recently introduced by Yuan and Zhang.

math.NT↗

arXiv2022

Lower bounds for Seshadri constants via successive minima of line bundles

Given a nef and big line bundle $L$ on a projective variety $X$ of dimension $d \geq 2$, we prove that the Seshadri constant of $L$ at a very general point is larger than $(d+1)^{\frac{1}{d}-1}$. This slightly improves the lower bound $1/d$ established by Ein, Küchle and Lazarsfeld. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito.

math.AG↗

arXiv2022

Arithmetic Okounkov bodies and positivity of adelic Cartier divisors

In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic $\mathbb{R}$-Cartier divisor $\overline{D}$ are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized $\mathbb{R}$-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of $\overline{D}$ coincides with the infimum of the Boucksom--Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.

math.AG↗

arXiv2020

Successive minima and asymptotic slopes in Arakelov Geometry

Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = \mathbb{P}_K^d$, our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Rémond.

math.AG↗