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

Abstract

The purpose of this paper is to introduce a notion of pairs of adelic $\mathbb{R}$-Cartier divisors and $\mathbb{R}$-base conditions, to define the arithmetic volumes of such pairs, and to establish fundamental positivity properties of such pairs. We show that the arithmetic volume of a pair has the Fujita approximation property and that the Gâteaux derivatives of the arithmetic volume function at a big pair along the directions of adelic $\mathbb{R}$-Cartier divisors are given by suitable arithmetic positive intersection numbers. As a consequence, we establish an Arakelov theoretic analogue of the classical Bonnesen--Diskant inequality in convex geometry.