Arithmetic positivity on toric varieties

Burgos Gil, José; Moriwaki, Atsushi; Philippon, Patrice; Sombra, Martín

doi:10.1090/jag/643

Authors: José Ignacio Burgos Gil, Atsushi Moriwaki, Patrice Philippon and Martín Sombra
Journal: J. Algebraic Geom. 25 (2016), 201-272
DOI: https://doi.org/10.1090/jag/643
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.

References

Sébastien Boucksom and Huayi Chen, Okounkov bodies of filtered linear series, Compos. Math. 147 (2011), no. 4, 1205–1229. MR 2822867, DOI https://doi.org/10.1112/S0010437X11005355
Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Solution to a non-Archimedean Monge-Ampère equation, J. Amer. Math. Soc. 28 (2015), no. 3, 617–667. MR 3327532, DOI https://doi.org/10.1090/S0894-0347-2014-00806-7
Yuri Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), no. 3, 465–476. MR 1470340, DOI https://doi.org/10.1215/S0012-7094-97-08921-3
José Ignacio Burgos Gil, Patrice Philippon, and Martín Sombra, Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque 360 (2014), vi+222 (English, with English and French summaries). MR 3222615
Matthew H. Baker and Robert Rumely, Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 625–688 (English, with English and French summaries). MR 2244226
Jose Ignacio Burgos, Arithmetic Chow rings and Deligne-Beilinson cohomology, J. Algebraic Geom. 6 (1997), no. 2, 335–377. MR 1489119
Antoine Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235 (French). MR 2244803, DOI https://doi.org/10.1515/CRELLE.2006.049
Huayi Chen, Arithmetic Fujita approximation, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 4, 555–578 (English, with English and French summaries). MR 2722508, DOI https://doi.org/10.24033/asens.2127
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322

A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique. I, Grunlehren der Math. Wissenschaften 166 (1971); II, Publ. Math. I.H.É.S. 8 (1961); III, ibid. 11 (1961); IV, ibid. 20 (1964), 24 (1965), 28 (1966), 32 (1967).

Charles Favre and Juan Rivera-Letelier, Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), no. 2, 311–361 (French, with English and French summaries). MR 2221116, DOI https://doi.org/10.1007/s00208-006-0751-x

William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037

Éric Gaudron, Géométrie des nombres adélique et lemmes de Siegel généralisés, Manuscripta Math. 130 (2009), no. 2, 159–182 (French, with English and French summaries). MR 2545512, DOI https://doi.org/10.1007/s00229-009-0282-3

Henri Gillet and Christophe Soulé, Amplitude arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 17, 887–890 (French, with English summary). MR 974432

Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93–174 (1991). MR 1087394

Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605

Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723

Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471

Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232

Atsushi Moriwaki, Continuity of volumes on arithmetic varieties, J. Algebraic Geom. 18 (2009), no. 3, 407–457. MR 2496453, DOI https://doi.org/10.1090/S1056-3911-08-00500-6

Atsushi Moriwaki, Big arithmetic divisors on the projective spaces over $\Bbb Z$, Kyoto J. Math. 51 (2011), no. 3, 503–534. MR 2823999, DOI https://doi.org/10.1215/21562261-1299882

Atsushi Moriwaki, Arithmetic linear series with base conditions, Math. Z. 272 (2012), no. 3-4, 1383–1401. MR 2995173, DOI https://doi.org/10.1007/s00209-012-0991-2

Atsushi Moriwaki, Zariski decompositions on arithmetic surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 4, 799–898. MR 2999543, DOI https://doi.org/10.2977/PRIMS/89

Atsushi Moriwaki, Toward Dirichlet’s unit theorem on arithmetic varieties, Kyoto J. Math. 53 (2013), no. 1, 197–259. MR 3049312, DOI https://doi.org/10.1215/21562261-1966116

Atsushi Moriwaki, Numerical characterization of nef arithmetic divisors on arithmetic surfaces, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 3, 717–753 (English, with English and French summaries). MR 3266711, DOI https://doi.org/10.5802/afst.1422

R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683

L. Szpiro, E. Ullmo, and S. Zhang, Équirépartition des petits points, Invent. Math. 127 (1997), no. 2, 337–347 (French). MR 1427622, DOI https://doi.org/10.1007/s002220050123

André Weil, Basic number theory, 3rd ed., Springer-Verlag, New York-Berlin, 1974. Die Grundlehren der Mathematischen Wissenschaften, Band 144. MR 0427267

Xinyi Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), no. 3, 603–649. MR 2425137, DOI https://doi.org/10.1007/s00222-008-0127-9

Xinyi Yuan, On volumes of arithmetic line bundles, Compos. Math. 145 (2009), no. 6, 1447–1464. MR 2575090, DOI https://doi.org/10.1112/S0010437X0900428X

Xinyi Yuan, On volumes of arithmetic line bundles. II, e-print arXiv:0909.3680v1, 2009.

Shouwu Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), no. 1, 187–221. MR 1254133, DOI https://doi.org/10.1090/S0894-0347-1995-1254133-7

Shouwu Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), no. 2, 281–300. MR 1311351

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
Email: burgos@icmat.es

Atsushi Moriwaki
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, 606-8502 Kyoto, Japan
Email: moriwaki@math.kyoto-u.ac.jp

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
Email: patrice.philippon@imj-prg.fr

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

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.
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