Heights on toric varieties for singular metrics: Global theory

TL;DR

This paper extends the theory of adelic divisors to toric varieties with singular metrics, providing a convex-analytic framework to compute arithmetic intersection numbers and heights in this setting.

Contribution

It develops a toric analog of adelic divisor theory and links arithmetic intersection numbers to integrals of concave functions, generalizing previous work to singular metrics.

Findings

Arithmetic self-intersection number equals the integral of a concave function.

Generalized intersection numbers match those by Burgos and Kramer (2024).

Framework enables computation of heights for toric varieties with singular metrics.

Abstract

In this paper, we develop a toric analog of the theory of adelic divisors on quasi-projective arithmetic varieties introduced by Yuan and Zhang, and extend the convex-analytic descriptions of the Arakelov geometry of projective toric arithmetic varieties given by Burgos, Philippon, and Sombra. Our main result is that the arithmetic self-intersection number of a semipositive toric adelic divisor is given by the integral of a concave function on a compact convex set. These generalized arithmetic intersection numbers coincide with the ones introduced by Burgos and Kramer in 2024, and therefore, can be used to compute heights of toric arithmetic varieties with respect to line bundles equipped with toric singular metrics.