Heights on toric varieties for singular metrics: Local theory

TL;DR

This paper generalizes the computation of local heights on toric varieties to include singular metrics, showing they can be expressed as integrals of concave functions over convex sets, extending previous results for continuous metrics.

Contribution

It extends the existing theory of local heights on toric varieties to singular metrics, providing a new integral formula involving concave functions.

Findings

Local height expressed as an integral over a convex set

Generalization from continuous to singular metrics

Answers an open question from 2016

Abstract

We show that the (toric) local height of a toric variety with respect to a semipositive torus-invariant singular metric is given by the integral of a concave function over a compact convex set. This generalizes a result of Burgos, Philippon, and Sombra for the case of continuous metrics and answers a question raised by Burgos, Kramer, and K\"uhn in 2016.