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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Arithmetic positivity on toric varieties

Authors: José Ignacio Burgos Gil, Atsushi Moriwaki, Patrice Philippon and Martín Sombra
Journal: J. Algebraic Geom. 25 (2016), 201-272
Published electronically: December 4, 2015
MathSciNet review: 3466351
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Abstract | References | Additional Information

Abstract: We continue the study of the arithmetic geometry of toric varieties started by J. Burgos Gil, P. Philippon, and M. Sombra in 2011. In this text, we study the positivity properties of metrized $\mathbb {R}$-divisors in the toric setting. For a toric metrized $\mathbb {R}$-divisor, we give formulae for its arithmetic volume and its $\chi$-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a higher-dimensional analogue of Dirichlet’s unit theorem for toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized $\mathbb {R}$-divisor, and we prove a toric arithmetic Fujita approximation theorem.

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Additional Information

José Ignacio Burgos Gil
Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3), Calle Nicolás Cabrera 15, Campus de la Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
MR Author ID: 349969

Atsushi Moriwaki
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, 606-8502 Kyoto, Japan

Patrice Philippon
Affiliation: Institut de Mathématiques de Jussieu – U.M.R. 7586 du CNRS, Équipe de Théorie des Nombres, Case 247, 4 place Jussieu, 75252 Paris cedex 05, France

Martín Sombra
Affiliation: ICREA and Universitat de Barcelona, Departament d’Àlgebra i Geometria, Gran Via 585, 08007 Barcelona, Spain
MR Author ID: 621582

Received by editor(s): October 26, 2012
Received by editor(s) in revised form: February 13, 2013
Published electronically: December 4, 2015
Additional Notes: The first author was partially supported by the MICINN research projects MTM2009-14163-C02-01 and MTM2010-17389. The second author was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (A) 22244003, 2011. The third author was partially supported by the CNRS international projects for scientific cooperation (PICS) “Properties of the heights of arithmetic varieties”, “Géométrie diophantienne et calcul formel”, and the ANR research project “Hauteur, modularité, transcendance”. The fourth author was partially supported by the MICINN research project MTM2009-14163-C02-01 and the MINECO research project MTM2012-38122-C03-02.
Article copyright: © Copyright 2015 University Press, Inc.