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

Abstract

We study the distribution of Galois orbits of points of small height on proper toric varieties, and its application to the Bogomolov problem. We introduce the notion of monocritical toric metrized divisor. We prove that a toric metrized divisor~$\overline{D}$ on a proper toric variety $X$ over a global field~$\Bbb{K}$ is monocritical if and only if for every generic $\overline{D}$-small sequence of algebraic points of $X$ and every place~$v$ of~$\Bbb{K}$, the sequence of their Galois orbits on the analytic space $X^{{\rm an}}_v$ converges to a measure. When this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of~$X$. The key ingredient is the study of the $v$-adic modulus distribution of Galois orbits of generic $\overline{D}$-small sequences of algebraic points. In particular, we characterize all their cluster measures. We generalize the Bogomolov problem by asking when a closed subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety, must be a translate of a subtorus. We prove that the generalized Bogomolov problem has a positive answer for monocritical toric metrized divisors, and we give several examples of toric metrized divisors for which the Bogomolov problem has a negative answer.