Volume function over a trivially valued field

TL;DR

This paper introduces an adelic Cartier divisor over a trivially valued field, explores its bigness, and establishes properties like continuity and log concavity of the arithmetic volume.

Contribution

It provides the first integral representation of the arithmetic volume and proves the existence of its limit, advancing the understanding of volume functions in this context.

Findings

Integral representation of arithmetic volume

Existence of limit of the volume

Continuity and log concavity of the volume

Abstract

We introduce an adelic Cartier divisor over a trivially valued field and discuss the bigness of it. For bigness, we give the integral representation of the arithmetic volume and prove the existence of limit of it. Moreover, we show that the arithmetic volume is continuous and log concave.