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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Arithmetic Okounkov bodies and positivity of adelic Cartier divisors

Author: François Ballaÿ
Journal: J. Algebraic Geom. 33 (2024), 455-492
Published electronically: October 17, 2023
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Abstract: In algebraic geometry, theorems of Küronya 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.

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François Ballaÿ
Affiliation: Université de Caen–Normandie, UNICAEN, CNRS, LMNO, 14000 Caen, France

Received by editor(s): June 17, 2022
Received by editor(s) in revised form: September 6, 2022
Published electronically: October 17, 2023
Article copyright: © Copyright 2023 University Press, Inc.