On the concavity of the arithmetic volumes

TL;DR

This paper investigates the differentiability of arithmetic volumes along arithmetic R-divisors and explores conditions for the Brunn-Minkowski inequality within the cone of nef and big arithmetic R-divisors.

Contribution

It provides new insights into the differentiability properties of arithmetic volumes and establishes equality conditions for the Brunn-Minkowski inequality in this context.

Findings

Arithmetic volume differentiability along R-divisors analyzed

Equality conditions for Brunn-Minkowski inequality identified

Results enhance understanding of arithmetic divisor geometry

Abstract

In this note, we study the differentiability of the arithmetic volumes along arithmetic R-divisors, and give some equality conditions for the Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big arithmetic R-divisors.