Continuous extension of arithmetic volumes

TL;DR

This paper extends the arithmetic volume function from rational to real coefficients, building on previous work that proved its continuity for smooth hermitian sheaves, enabling broader applications.

Contribution

It introduces a continuous extension of the arithmetic volume function over real numbers, expanding the scope of previous continuity results for hermitian Q-invertible sheaves.

Findings

Extended the volume function to real coefficients

Proved the continuity of the extended volume function

Enabled new applications in arithmetic geometry

Abstract

This paper is the sequel of the paper "Continuity of volumes on arithmetic varieties", in which we established the arithmetic volume function of smooth hermitian Q-invertible sheaves and proved its continuity. The continuity of the volume function has a lot of applications as treated in the paper as above. In this paper, we would like to consider its continuous extension over R.