Remarks on the arithmetic restricted volumes and the arithmetic base loci

TL;DR

This paper explores fundamental properties of arithmetic restricted volumes of adelically metrized line bundles, highlighting their concavity, relation to base loci, and a generalized Fujita approximation.

Contribution

It establishes key properties of arithmetic restricted volumes, including concavity and their role in characterizing base loci, along with a generalized Fujita approximation.

Findings

Arithmetic restricted volume has concavity property.

Characterizes the arithmetic augmented base locus as the null locus.

Provides a generalized Fujita approximation for these volumes.

Abstract

In this paper, we collect some fundamental properties of the arithmetic restricted volumes (or the arithmetic multiplicities) of the adelically metrized line bundles. The arithmetic restricted volume has the concavity property and characterizes the arithmetic augmented base locus as the null locus. We also show a generalized Fujita approximation for the arithmetic restricted volumes.