Tohoku Mathematical Journal
2021

September
SECOND SERIES VOL. 73, NO. 3

Tohoku Math. J.
73 (2021), 341-401

Title ADELIC CARTIER DIVISORS WITH BASE CONDITIONS AND THE BONNESEN--DISKANT-TYPE INEQUALITIES

Author Hideaki Ikoma

(Received December 14, 2018, revised April 1, 2020)
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.

Mathematics Subject Classification. Primary 14G40; Secondary 11G50.

Key words and phrases. Arakelov theory, adelic divisors, arithmetic volumes, differentiability, Bonnesen--Diskant inequality.

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