The distribution of Galois orbits of points of small height in toric varieties

Original Title: “The distribution of Galois orbits of points of small height in toric varieties”

Authors: José IgnacioBurgos Gil (ICMAT-CSIC), Patrice Philippon (Institut de Mathematiques de Jussieu – U.M.R. 7586 du CNRS), Juan RiveraLetelier (University of Rochester) y Martín Sombra (ICREA, UB)

Source: American Journal of Mathematics 141 (2019), 309381

Date of online publication: 2019



Equidistribution results are an important tool in many branches of mathematics. For instance, the equidistribution property the Galois orbits of small points in abelian varieties over number fields was studied by L. Szpiro, E. Ullmo and S. Zhang and led to a proof of the Bogomolov conjecture. This equidistribution property has been widely generalized. In particular, it has been extended to more general varieties and height functions and, with the introduction of Berkovich spaces, to non-Archimedean places. Up to the date of this paper, all these generalizations are restricted to height functions that satisfy a special condition, namely, that the essential minimum of the heights of points is equal to the normalized height of the ambient variety. Many height functions satisfy this special condition, like Néron-Tate heights on Abelian varieties, canonical metrics on toric varieties, and more generally those coming from algebraic dynamical systems. But there are also many height functions of interest that do not satisfy it, like (twisted) Fubini-Study heights on projective spaces and the Faltings height on modular varieties.

Toric varieties form a very nice class of varieties that have a combinatorial description and allow very explicit computations. This paper contains a complete study of the equidistribution property of Galois orbits of small points with respect to toric height functions and the associated Bogomolov property. The authors show that a mild positivity assumption is enough to guarantee equidistribution. This provides a wealth of new height functions for which the equidistribution property holds. Moreover, they give a complete classification of those toric heights for which equidistribution holds, and they use it to prove that the equidistribution property implies the Bogomolov property in the toric context. As a byproduct, they give a characterization of those toric heights whose essential minimum is attained. They also provide examples of toric height functions that do not have the Bogomolov property and for which the equidistribution property fails in a myriad of ways.