Arithmetic positivity on toric varieties

TL;DR

This paper advances the understanding of positivity properties of metrized divisors on toric varieties, providing explicit formulas and characterizations, and applying these to fundamental theorems in arithmetic geometry.

Contribution

It introduces formulas for arithmetic volumes of toric metrized divisors and characterizes positivity notions using combinatorial data, along with applications to Dirichlet's theorem and Fujita approximation.

Findings

Formulas for arithmetic and chi-arithmetic volumes of toric divisors

Characterization of arithmetically ample, nef, big, and pseudo-effective divisors

Proof of a toric arithmetic Fujita approximation theorem

Abstract

We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor, we give formulae for its arithmetic volume and its chi-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a Dirichlet's unit theorem on toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R-divisor, and we prove a toric arithmetic Fujita approximation theorem.