Metrized ample line bundles in non-Archimedean geometry II

TL;DR

This paper introduces Shilov finite metrics on ample line bundles in non-Archimedean geometry, providing new tools for calculating metric distortions and proving an arithmetic Hilbert-Samuel formula, along with defining equidistribution measures.

Contribution

It defines Shilov finite metrics, computes determinant metric distortions, and proves an arithmetic Hilbert-Samuel formula using non-Archimedean techniques.

Findings

Introduction of Shilov finite metrics

Calculation of determinant metric distortion

Analytic proof of arithmetic Hilbert-Samuel formula

Abstract

We introduce a class of semipositive metrics on ample line bundles in non-Archimedean geometry, called Shilov finite metrics. We calculate the determinant metric distorsion in the exact sequence induced by a global section using non-Archimedean norm reduction techniques. This leads to an analytic proof to the arithmetic Hilbert-Samuel formula over a local place for a semipositively metrized ample line bundle. We define the equidistribution measure associated to a Shilov finite metric and solve the corresponding inverse problem.