Volumes of Arithmetic Okounkov Bodies

TL;DR

This paper establishes that the volume of the arithmetic Okounkov body, derived from a hermitian line bundle on an arithmetic variety, is proportional to the volume of the line bundle, simplifying previous results.

Contribution

It proves the volume equality between the arithmetic Okounkov body and the hermitian line bundle, improving and simplifying earlier work.

Findings

Volume of arithmetic Okounkov body equals the line bundle volume up to a constant

Simplifies previous proofs of the volume relation

Provides a clearer understanding of the geometric properties of arithmetic line bundles

Abstract

This paper proves the volume of the arithmetic Okounkov body, constructed from a hermitian line bundle on an arithmetic variety by the author in a previous paper, is equal to the the volume of the hermitian line bundle up to a simple constant multiple. It is an improvement and simplification of the previous work.