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Local Heights of Toric Varieties over Non-archimedean Fields
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2017), pp. 5-77.

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.

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.

Published online:
DOI: 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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Local {Heights} of {Toric} {Varieties} over {Non-archimedean} {Fields}},
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     year = {2017},
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Walter Gubler; Julius Hertel. Local Heights of Toric Varieties over Non-archimedean Fields. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2017), pp. 5-77. doi : 10.5802/pmb.15. https://pmb.centre-mersenne.org/articles/10.5802/pmb.15/

[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 | Zbl

[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 | DOI | Numdam | MR | Zbl

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

[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 | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

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

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

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

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

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

[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

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

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

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

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

[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 | Zbl

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

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

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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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