Toric metrics on a line bundle of an abelian variety are the invariant metrics under the natural torus action coming from Raynaud’s uniformization theory. We compute here the associated Monge–Ampère measures for the restriction to any closed subvariety of . This generalizes the computation of canonical measures done by the first author from canonical metrics to toric metrics and from discrete valuations to arbitrary non-archimedean fields.
Les métriques toriques sur un fibré en droites sur une variété abélienne sont les métriques invariantes sous l’action naturelle du tore issue de la théorie de l’uniformisation de Raynaud. Nous calculons les mesures de Monge–Ampère associées pour les restrictions à toutes les sous-variétés fermées de . Ceci généralise des travaux du premier auteur sur le calcul des mesures canoniques pour des valuations discrètes au cas des métriques toriques pour des corps non archimédiens arbitraires.
Mots-clés : Berkovich analytic spaces, formal geometry, abelian varieties, canonical measures
Walter Gubler 1; Stefan Stadlöder 1

@article{PMB_2023____49_0, author = {Walter Gubler and Stefan Stadl\"oder}, title = {Monge{\textendash}Amp\`ere measures for toric metrics on abelian varieties}, journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres}, pages = {49--84}, publisher = {Presses universitaires de Franche-Comt\'e}, year = {2023}, doi = {10.5802/pmb.49}, language = {en}, url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.49/} }
TY - JOUR AU - Walter Gubler AU - Stefan Stadlöder TI - Monge–Ampère measures for toric metrics on abelian varieties JO - Publications mathématiques de Besançon. Algèbre et théorie des nombres PY - 2023 SP - 49 EP - 84 PB - Presses universitaires de Franche-Comté UR - https://pmb.centre-mersenne.org/articles/10.5802/pmb.49/ DO - 10.5802/pmb.49 LA - en ID - PMB_2023____49_0 ER -
%0 Journal Article %A Walter Gubler %A Stefan Stadlöder %T Monge–Ampère measures for toric metrics on abelian varieties %J Publications mathématiques de Besançon. Algèbre et théorie des nombres %D 2023 %P 49-84 %I Presses universitaires de Franche-Comté %U https://pmb.centre-mersenne.org/articles/10.5802/pmb.49/ %R 10.5802/pmb.49 %G en %F PMB_2023____49_0
Walter Gubler; Stefan Stadlöder. Monge–Ampère measures for toric metrics on abelian varieties. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 49-84. doi : 10.5802/pmb.49. https://pmb.centre-mersenne.org/articles/10.5802/pmb.49/
[1] Log smoothness and polystability over valuation rings (2019) (http://arxiv.org/abs/1806.09168)
[2] Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990, x+169 pages | MR
[3] Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 78 (1993), pp. 5-161 | DOI | Numdam | MR | Zbl
[4] Smooth -adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | DOI | MR | Zbl
[5] Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006, xvi+652 pages | DOI | MR
[6] Lectures on formal and rigid geometry, Lecture Notes in Mathematics, 2105, Springer, 2014, viii+254 pages | DOI
[7] Degenerating abelian varieties, Topology, Volume 30 (1991) no. 4, pp. 653-698 | DOI | MR | Zbl
[8] A comparison of positivity in complex and tropical toric geometry, Math. Z., Volume 299 (2021) no. 3-4, pp. 1199-1255 | DOI | MR | Zbl
[9] Pluripotential theory for tropical toric varieties and non-archimedean Monge-Ampère equations (2021) (https://arxiv.org/abs/2102.07392)
[10] Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, 360, Société Mathématique de France, 2014, vi+222 pages | Numdam | MR | Zbl
[11] The geometric Bogomolov conjecture, Duke Math. J., Volume 170 (2021) no. 2, pp. 247-277 | DOI | MR | Zbl
[12] Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | DOI | MR | Zbl
[13] Formes différentielles réelles et courants sur les espaces de Berkovich (2012) (http://arxiv.org/abs/1204.6277)
[14] Espaces de Berkovich, polytopes, squelettes et théorie des modèles, Confluentes Math., Volume 4 (2012) no. 4, 1250007, 57 pages | DOI | MR | Zbl
[15] Families of Berkovich spaces, Astérisque, 400, Société Mathématique de France, 2018, vii+262 pages | Zbl
[16] Diophantine approximation on abelian varieties, Ann. Math., Volume 133 (1991) no. 3, pp. 549-576 | DOI | MR | Zbl
[17] Non-Archimedean and tropical theta functions, Math. Ann., Volume 372 (2018) no. 3-4, pp. 891-914 | DOI | MR | Zbl
[18] Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, 1993, xii+157 pages | DOI | MR
[19] Heights in families of abelian varieties and the geometric Bogomolov conjecture, Ann. Math., Volume 189 (2019) no. 2, pp. 527-604 | DOI | MR | Zbl
[20] Local heights of subvarieties over non-Archimedean fields, J. Reine Angew. Math., Volume 498 (1998), pp. 61-113 | DOI | MR | Zbl
[21] The Bogomolov conjecture for totally degenerate abelian varieties, Invent. Math., Volume 169 (2007) no. 2, pp. 377-400 | DOI | MR | Zbl
[22] Tropical varieties for non-Archimedean analytic spaces, Invent. Math., Volume 169 (2007) no. 2, pp. 321-376 | DOI | MR | Zbl
[23] Non-Archimedean canonical measures on abelian varieties, Compos. Math., Volume 146 (2010) no. 3, pp. 683-730 | DOI | MR | Zbl
[24] A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry (Contemporary Mathematics), Volume 589, American Mathematical Society, 2013, pp. 125-189 | DOI | MR | Zbl
[25] Forms on Berkovich spaces based on harmonic tropicalizations (2021) (https://arxiv.org/abs/1909.12633)
[26] A tropical approach to nonarchimedean Arakelov geometry, Algebra Number Theory, Volume 11 (2017) no. 1, pp. 77-180 | DOI | MR | Zbl
[27] On Zhang’s semipositive metrics, Doc. Math., Volume 24 (2019), pp. 331-372 | DOI | MR | Zbl
[28] The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic, Volume 112 (2001) no. 1, pp. 43-115 | DOI | MR | Zbl
[29] Super currents and tropical geometry, Math. Z., Volume 270 (2012) no. 3-4, pp. 1011-1050 | DOI | MR | Zbl
[30] Tropical curves, their Jacobians and theta functions, Curves and abelian varieties (Contemporary Mathematics), Volume 465, American Mathematical Society, 2008, pp. 203-230 | DOI | MR | Zbl
[31] On -invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, J. Algebr. Geom., Volume 13 (2004) no. 4, pp. 771-798 | DOI | MR | Zbl
[32] Variétés abéliennes et géométrie rigide, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, 1971, pp. 473-477 | MR | Zbl
[33] Sous-variétés d’une variété abélienne et points de torsion, Arithmetic and geometry, Vol. I (Progress in Mathematics), Volume 35, Birkhäuser, 1983, pp. 327-352 | DOI | MR | Zbl
[34] Canonical measures of subvarieties of abelian varieties, Ph. D. Thesis, Universität Regensburg (2022) (http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:355-epub-526463)
[35] Équirépartition des petits points, Invent. Math., Volume 127 (1997) no. 2, pp. 337-347 | DOI | MR | Zbl
[36] Positivité et discrétion des points algébriques des courbes, Ann. Math., Volume 147 (1998) no. 1, pp. 167-179 | DOI | MR | Zbl
[37] Geometric Bogomolov conjecture in arbitrary characteristics (2021) (http://arxiv.org/abs/2108.09722)
[38] Big line bundles over arithmetic varieties, Invent. Math., Volume 173 (2008) no. 3, pp. 603-649 | DOI | MR | Zbl
[39] Equidistribution of small points on abelian varieties, Ann. Math., Volume 147 (1998) no. 1, pp. 159-165 | DOI | MR | Zbl
Cited by Sources: