Arithmetic Okounkov bodies and positivity of adelic Cartier divisors

TL;DR

This paper extends classical geometric criteria for divisor positivity to Arakelov geometry using arithmetic Okounkov bodies, providing new tools for understanding adelic divisors and their arithmetic properties.

Contribution

It generalizes positivity criteria for divisors from classical to arithmetic settings via Okounkov bodies, including applications to small points and subvarieties.

Findings

Arithmetic ampleness and nefness characterized by arithmetic Okounkov bodies.

Established the equality of the absolute minimum and Boucksom--Chen transform.

Proved a converse to the arithmetic Hilbert-Samuel theorem.

Abstract

In algebraic geometry, theorems of K\"uronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic $\mathbb{R}$-Cartier divisor $\overline{D}$ are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized $\mathbb{R}$-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of $\overline{D}$ coincides with the infimum of the Boucksom--Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.