Adelic Cartier divisors with base conditions and the Bonnesen-Diskant-type inequalities

TL;DR

This paper develops positivity concepts for adelic Cartier divisors with base conditions, studies their arithmetic volumes, and derives an Arakelov-theoretic analogue of the Bonnesen-Diskant inequality, connecting arithmetic geometry and convex geometry.

Contribution

It introduces new positivity notions for adelic divisors with base conditions and establishes a Bonnesen-Diskant-type inequality in Arakelov geometry.

Findings

Arithmetic volume derivatives are given by positive intersection numbers.

Established an Arakelov analogue of the Bonnesen-Diskant inequality.

Provided fundamental properties of arithmetic volumes for pairs of adelic divisors.

Abstract

In this paper, we introduce positivity notions for pairs of adelic R-Cartier divisors and R-base conditions, and study fundamental properties of the arithmetic volumes defined for such pairs. We show that the Gateaux derivatives of the arithmetic volume function at big pairs along the directions of adelic R-Cartier divisors are given by suitable arithmetic positive intersection numbers. As a corollary, we obtain an Arakelov theoretic analogue of the Bonnesen-Diskant inequality in convex geometry.