Local Heights of Toric Varieties over Non-archimedean Fields
[Hauteurs locales des variétés toriques sur un corps ultramétrique complet]
Publications Mathématiques de Besançon (2017), pp. 5-77.

Nous généralisons des résultats concernant les hauteurs locales prouvés précédemment pour une valuation discrète au cas d’une valeur absolue ultramétrique quelconque. Nous traitons tout d’abord le case de la formule de récurrence de Chambert-Loir et Thuillier. Ensuite nous généralisons la formule de Burgos–Philippon–Sombra pour la hauteur locale torique d’une variété torique normale propre. Nous appliquons la formule correspondante de Moriwaki pour les hauteurs globales sur un corps de type fini au cas d’une fibration qui est génériquement torique. Nous illustrons ce dernier résultat par un exemple naturel où des valuations non discrètes jouent un rôle important.

We generalize results about local heights previously proved in the case of discrete absolute values to arbitrary non-archimedean absolute values. First, this is done for the induction formula of Chambert-Loir and Thuillier. Then we prove the formula of Burgos–Philippon–Sombra for the toric local height of a proper normal toric variety in this more general setting. We apply the corresponding formula for Moriwaki’s global heights over a finitely generated field to a fibration which is generically toric. We illustrate the last result in a natural example where non-discrete non-archimedean absolute values really matter.

Publié le : 2017-11-30
DOI : https://doi.org/10.5802/pmb.15
Classification : 14M25,  14G40,  14G22
Mots clés: Toric geometry, local heights, berkovich spaces, Chambert-Loir measure, heights of varieties over finitely generated fields
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     author = {Walter Gubler and Julius Hertel},
     title = {Local Heights of Toric Varieties over Non-archimedean Fields},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2017},
     pages = {5-77},
     doi = {10.5802/pmb.15},
     language = {en},
     url = {pmb.centre-mersenne.org/item/PMB_2017____5_0/}
}
Walter Gubler; Julius Hertel. Local Heights of Toric Varieties over Non-archimedean Fields. Publications Mathématiques de Besançon (2017), pp. 5-77. doi : 10.5802/pmb.15. https://pmb.centre-mersenne.org/item/PMB_2017____5_0/

[1] Vladimir G. Berkovich Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, Volume 33, American Mathematical Society, 1990, ix+169 pages | MR 1070709 | Zbl 0715.14013

[2] Vladimir G. Berkovich Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 78 (1993) no. 78, pp. 5-161 | Article | Numdam | MR 1259429 | Zbl 0804.32019

[3] Vladimir G. Berkovich Smooth p-adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | Article | MR 1702143 | Zbl 0930.32016

[4] Vladimir G. Berkovich Smooth p-adic analytic spaces are locally contractible. II, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, 2004, pp. 293-370 | MR 2023293 | Zbl 1060.32010

[5] Enrico Bombieri; Walter Gubler Heights in Diophantine Geometry, New Mathematical Monographs, Volume 4, Cambridge University Press, 2006, xvi+652 pages | MR 2216774 | Zbl 1115.11034

[6] Siegfried Bosch; Ulrich Güntzer; Reinhold Remmert Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, Volume 261, Springer, 1984, xii+436 pages | MR 746961 | Zbl 0539.14017

[7] Siegfried Bosch; Werner Lütkebohmert Stable reduction and uniformization of abelian varieties. I, Math. Ann., Volume 270 (1985) no. 3, pp. 349-379 | Article | MR 774362 | Zbl 0554.14012

[8] Siegfried Bosch; Werner Lütkebohmert Formal and rigid geometry. II. Flattening techniques, Math. Ann., Volume 296 (1993) no. 3, pp. 403-429 | Article | MR 1225983 | Zbl 0808.14018

[9] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math., Volume 119 (1995) no. 2, pp. 361-398 | Article | MR 1312505 | Zbl 0839.14014

[10] Jean-Benoît Bost; Henri Gillet; Christophe Soulé Heights of projective varieties and positive Green forms, J. Am. Math. Soc., Volume 7 (1994) no. 4, pp. 903-1027 | Article | MR 1260106 | Zbl 0973.14013

[11] José Ignacio Burgos Gil; Patrice Philippon; Martín Sombra Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, Volume 360, Société Mathématique de France, 2014, vi+222 pages | Zbl 1311.14050

[12] José Ignacio Burgos Gil; Patrice Philippon; Martín Sombra Height of varieties over finitely generated fields, Kyoto J. Math., Volume 56 (2016) no. 1, pp. 13-32 | Article | MR 3479316 | Zbl 1358.14021

[13] José Ignacio Burgos Gil; Martín Sombra When do the recession cones of a polyhedral complex form a fan?, Discrete Comput. Geom., Volume 46 (2011) no. 4, pp. 789-798 | Article | MR 2846179 | Zbl 1233.14031

[14] Antoine Chambert-Loir Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl 1112.14022

[15] Antoine Chambert-Loir; Amaury Thuillier Mesures de Mahler et équidistribution logarithmique, Ann. Inst. Fourier, Volume 59 (2009) no. 3, pp. 977-1014 | Article | Numdam | Zbl 1192.14020

[16] David A. Cox; John B. Little; Henry K. Schenck Toric Varieties, Graduate studies in Mathematics, Volume 124, American Mathematical Society, 2011, xxiv+841 pages | MR 2810322 | Zbl 1223.14001

[17] Antoine Ducros Les espaces de Berkovich sont excellents, Ann. Inst. Fourier, Volume 59 (2009) no. 4, pp. 1443-1552 | Article | Numdam | MR 2566967 | Zbl 1177.14049

[18] Gerd Faltings Diophantine approximation on abelian varieties, Ann. Math., Volume 133 (1991) no. 3, pp. 549-576 | Article | MR 1109353 | Zbl 0734.14007

[19] William Fulton Introduction to Toric Varieties, Annals of mathematics studies, Volume 131, Princeton University Press, 1993, xi+157 pages | MR 1234037 | Zbl 0813.14039

[20] William Fulton Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, Volume 2, Springer, 1998, xiii+470 pages | MR 1644323 | Zbl 0885.14002

[21] Henri Gillet; Christophe Soulé Arithmetic intersection theory, Publ. Math., Inst. Hautes Étud. Sci., Volume 72 (1990) no. 72, pp. 93-174 | Numdam | MR 1087394 | Zbl 0741.14012

[22] Ulrich Görtz; Torsten Wedhorn Algebraic Geometry I: Schemes With Examples and Exercises, Advanced Lectures in Mathematics, Vieweg+Teubner Verlag, 2010, vii+615 pages | Zbl 1213.14001

[23] Alexander Grothendieck Éléments de géométrie algébrique: I. Le langage des schémas, Publ. Math., Inst. Hautes Étud. Sci., Volume 4 (1960), pp. 5-228 | Article | Numdam | Zbl 0118.36206

[24] Walter Gubler Heights of subvarieties over M-fields, Arithmetic geometry (Cortona, 1994) (Sympos. Math.) Volume 37 (1997), pp. 190-227 | MR 1472498 | Zbl 0916.14011

[25] Walter Gubler Local heights of subvarieties over non-Archimedean fields, J. Reine Angew. Math., Volume 498 (1998), pp. 61-113 | Article | MR 1629925 | Zbl 0906.14013

[26] Walter Gubler Basic Properties of Heights of Subvarieties (2002) (Ph. D. Thesis) | Article

[27] Walter Gubler Local and Canonical Heights of Subvarieties, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 2 (2003) no. 4, pp. 711-760 | Numdam | MR 2040641 | Zbl 1170.14303

[28] Walter Gubler The Bogomolov conjecture for totally degenerate abelian varieties, Invent. Math., Volume 169 (2007) no. 2, pp. 377-400 | Article | MR 2318560 | Zbl 1153.14029

[29] Walter Gubler Tropical varieties for non-Archimedean analytic spaces, Invent. Math., Volume 169 (2007) no. 2, pp. 321-376 | Article | MR 2318559 | Zbl 1153.14036

[30] Walter Gubler Equidistribution over function fields, Manuscr. Math., Volume 127 (2008) no. 4, pp. 485-510 | Article | MR 2457191 | Zbl 1189.14030

[31] Walter Gubler Non-Archimedean canonical measures on abelian varieties, Compos. Math., Volume 146 (2010) no. 3, pp. 683-730 | Article | MR 2644932 | Zbl 1192.14021

[32] Walter Gubler A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry (Contemporary Mathematics) Volume 589 (2013), pp. 125-189 | Article | MR 3088913 | Zbl 1318.14061

[33] Walter Gubler; Kuennemann Künnemann Positivity properties of metrics and delta-forms (2015) (https://arxiv.org/abs/1509.09079v1)

[34] Walter Gubler; Joseph Rabinoff; Annette Werner Skeletons and tropicalizations, Adv. Math., Volume 294 (2016), pp. 150-215 | Article | MR 3479562 | Zbl 06567870

[35] Walter Gubler; Alejandro Soto Classification of normal toric varieties over a valuation ring of rank one, Doc. Math., J. DMV, Volume 20 (2015), pp. 171-198 | MR 3398711 | Zbl 1349.14161

[36] Takeshi Kajiwara Tropical toric geometry, Toric topology (Contemporary Mathematics) Volume 460, American Mathematical Society, 2008, pp. 197-207 | Article | MR 2428356 | Zbl 1202.14047

[37] Eric Katz; Joseph Rabinoff; David Zureick-Brown Uniform bounds for the number of rational points on curves of small Mordell–Weil rank, Duke Math. J., Volume 165 (2016) no. 16, pp. 3189-3240 | Article | MR 3566201 | Zbl 06666955

[38] George R. Kempf; Finn Faye Knudsen; David Bryant Mumford; Bernard Saint-Donat Toroidal Embeddings I, Lecture Notes in Mathematics, Volume 339, Springer, 1973, viii+209 pages | MR 335518 | Zbl 0271.14017

[39] Steven Lawrence Kleiman Toward a numerical theory of ampleness, Ann. Math., Volume 84 (1966), pp. 293-344 | Article | MR 206009 | Zbl 0146.17001

[40] Atsushi Moriwaki Arithmetic height functions over finitely generated fields, Invent. Math., Volume 140 (2000) no. 1, pp. 101-142 | Article | MR 1779799 | Zbl 1007.11042

[41] Sam Payne Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 543-556 | Article | MR 2511632 | Zbl 1193.14077

[42] Ralph Tyrrell Rockafellar Convex analysis, Princeton Mathematical Series, Volume 28, Princeton University Press, 1970, xviii+451 pages | MR 274683 | Zbl 0193.18401

[43] Stacks Project Authors Stacks Project, 2015 (http://stacks.math.columbia.edu)

[44] Shou-Wu Zhang Small points and adelic metrics, J. Algebr. Geom., Volume 4 (1995) no. 2, pp. 281-300 | MR 1311351 | Zbl 0861.14019

[45] Shou-Wu Zhang Equidistribution of small points on abelian varieties, Ann. Math., Volume 147 (1998) no. 1, pp. 159-165 | Article | MR 1609518 | Zbl 0991.11034