On Volumes of Arithmetic Line Bundles

TL;DR

This paper extends the theory of volumes of line bundles to the arithmetic setting, introducing new inequalities and approximation theorems that generalize classical geometric results using Okounkov bodies.

Contribution

It provides an arithmetic generalization of Lazarsfeld-Mustata's work, establishing log-concavity and Fujita approximation for arithmetic line bundles.

Findings

Established a log-concavity inequality for volumes of arithmetic line bundles.

Proved an arithmetic Fujita approximation theorem for big line bundles.

Extended the use of Okounkov bodies to the arithmetic context.

Abstract

We show an arithmetic generalization of the recent work of Lazarsfeld-Mustata which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.