Arakelov geometry of toric bundles: Okounkov bodies and BKK

TL;DR

This paper explores the arithmetic geometry of toric bundles, focusing on Okounkov bodies and intersection theory, and extends existing height and minima computations to broader toric compactifications.

Contribution

It introduces the study of adelic line bundles on toric bundles and proves an arithmetic analogue of a key intersection number formula.

Findings

Derived formulas for intersection numbers of adelic line bundles.

Computed heights and successive minima for compactifications of semiabelian varieties.

Extended Chambert-Loir's computations to arbitrary toric compactifications.

Abstract

This article introduces the study of toric bundles and the morphisms between them from the perspective of adelic fibre bundles, as introduced by Chambert-Loir and Tschinkel. We study the Okounkov bodies and Boucksom-Chen transforms of suitable adelic line bundles on toric bundles. Finally, we prove an arithmetic analogue of a formula for intersection numbers due to Hofscheier, Khovanskii and Monin. We apply this to the study of compactifications of semiabelian varieties, whose height and successive minima we compute. This extends computations of Chambert-Loir to arbitrary toric compactifications.