no. 4
Higher dimensional essential minima and equidistribution of cycles
[Minimums essentiels de dimension supérieure et équirépartition de cycles]
Gualdi, Roberto1 ; Martínez, César1
1 Fakultät für Mathematik Universität Regensburg 93040 Regensburg (Germany)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1329-1377.

Le minimum essentiel et l’équirépartition des petits points sont deux sujets entrelacés et bien établis en géométrie arithmétique. Cependant, on manque d’un analogue pour le minimum essentiel pour des sous-variétés de dimension supérieure, et l’équirépartition de celles-ci est un sujet beaucoup moins exploré.

Dans cet article, on introduit une nouvelle notion de minimum essentiel de dimension supérieure et on l’utilise pour prouver un résultat d’équirépartition pour des cycles effectifs. Celui-ci généralise les théorèmes d’équirépartition de dimension supérieure précédents en permettant plus de flexibilité sur les données arithmétiques.

The essential minimum and equidistribution of small points are two well-established interrelated subjects in arithmetic geometry. However, there is lack of an analogue of essential minimum dealing with higher dimensional subvarieties, and the equidistribution of these is a far less explored topic.

In this paper, we introduce a new notion of higher dimensional essential minimum and use it to prove equidistribution of generic and small effective cycles. The latter generalizes the previous higher dimensional equidistribution theorems by considering cycles and by allowing more flexibility on the arithmetic datum.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3500
Classification : 14G40, 11G35, 14C25
Keywords: Equidistribution of cycles, Arakelov geometry, Heights, Essential minimum
Mot clés : Équirépartition de cycles, Géométrie d’Arakelov, Hauteurs, Minimum essentiel
Gualdi, Roberto 1 ; Martínez, César 1

1 Fakultät für Mathematik Universität Regensburg 93040 Regensburg (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Gualdi, Roberto; Martínez, César. Higher dimensional essential minima and equidistribution of cycles. Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1329-1377. doi : 10.5802/aif.3500. https://aif.centre-mersenne.org/articles/10.5802/aif.3500/

[1] Autissier, Pascal Équidistribution des sous-variétés de petite hauteur, J. Théor. Nombres Bordeaux, Volume 18 (2006), pp. 1-12 | DOI | Numdam | MR | Zbl

[2] Baker, Matthew; Ih, Su-ion Equidistribution of small subvarieties of an abelian variety, New York J. Math., Volume 10 (2004), pp. 279-285 | MR | Zbl

[3] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990, x+169 pages | MR

[4] Bilu, Yuri Limit distribution of small points on algebraic tori, Duke Math. J., Volume 89 (1997) no. 3, pp. 465-476 | DOI | MR | Zbl

[5] Bombieri, Enrico; Gubler, Walter Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006

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

[7] Burgos Gil, José Ignacio; Philippon, Patrice; Rivera-Letelier, Juan; Sombra, Martín The distribution of Galois orbits of points of small height in toric varieties, Am. J. Math., Volume 141 (2019) no. 2, pp. 309-381 | MR | Zbl

[8] Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, 360, Société Mathématique de France, 2014

[9] Chambert-Loir, Antoine Géométrie d’Arakelov et hauteurs canoniques sur des variétés semi-abéliennes, Math. Ann., Volume 314 (1999) no. 2, pp. 381-401 | Zbl

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

[11] Chambert-Loir, Antoine Heights and measures on analytic spaces. A survey of recent results, and some remarks, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II (London Mathematical Society Lecture Note Series), Volume 384, Cambridge University Press, 2011, pp. 1-50 | MR | Zbl

[12] Chambert-Loir, Antoine Arakelov geometry, heights, equidistribution, and the Bogomolov conjecture, Arakelov geometry and diophantine applications (Lecture Notes in Mathematics), Volume 2276, Springer, 2021, pp. 299-328 | DOI | MR | Zbl

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

[14] Chen, Huayi Positive degree and arithmetic bigness (2008) (https://arxiv.org/abs/0803.2583)

[15] Demailly, Jean-Pierre Monge–Ampère operators, Lelong numbers and intersection theory, Complex analysis and geometry (The University Series in Mathematics), Plenum Press, 1993, pp. 115-193 | DOI | Zbl

[16] Dujardin, Romain Some problems of arithmetic origin in rational dynamics, Arakelov geometry and diophantine applications (Lecture Notes in Mathematics), Volume 2276, Springer, 2021, pp. 341-374 | DOI | MR | Zbl

[17] Favre, Charles; Rivera-Letelier, Juan Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., Volume 335 (2006) no. 2, pp. 311-361 | DOI | Zbl

[18] Gubler, Walter Heights of subvarieties over M-fields, Arithmetic geometry (Cortona, 1994) (Symposia Mathematica), Volume 34, Cambridge University Press, 1997, pp. 190-227 | MR | Zbl

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

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

[21] Gubler, Walter Equidistribution over function fields, Manuscr. Math., Volume 127 (2008), pp. 485-510 | DOI | MR | Zbl

[22] Gubler, Walter; Hertel, Julius Local heights of toric varieties over non-archimedean fields, Actes de la Conférence “Non-Archimedean Analytic Geometry: Theory and Practice” (Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres), Volume 2017/1, Presses Universitaires de Franche-Comté, 2017, pp. 5-77 | Numdam | MR | Zbl

[23] Gubler, Walter; Kuennemann, Klaus Positivity properties of metrics and delta-forms, J. Reine Angew. Math., Volume 752 (2019), pp. 141-177 | DOI | MR | Zbl

[24] Gubler, Walter; Künnemann, Klaus A tropical approach to nonarchimedean Arakelov geometry, Algebra Number Theory, Volume 11 (2017) no. 1, pp. 77-180 | DOI | MR | Zbl

[25] Gubler, Walter; Martin, Florent On Zhang’s semipositive metrics, Doc. Math., Volume 24 (2019), pp. 331-372 | MR | Zbl

[26] Kühne, Lars Points of Small Height on Semiabelian Varieties, J. Eur. Math. Soc., Volume 24 (2022) no. 6, pp. 2077-2131 | DOI | MR | Zbl

[27] Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004

[28] Maillot, Vincent Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, Mémoires de la Société Mathématique de France, 80, Société Mathématique de France, 2000 | MR

[29] Martínez, César; Sombra, Martín An arithmetic Bernštein–Kušnirenko inequality, Math. Z., Volume 291 (2019) no. 3-4, pp. 1211-1244 | DOI | Zbl

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

[31] Moriwaki, Atsushi Continuity of volumes on arithmetic varieties, J. Algebr. Geom., Volume 18 (2009) no. 3, pp. 407-457 | DOI | MR | Zbl

[32] Moriwaki, Atsushi Continuous extension of arithmetic volumes, Int. Math. Res. Not., Volume 2009 (2009) no. 19, pp. 3598-3638 | DOI | MR | Zbl

[33] Moriwaki, Atsushi Arakelov geometry, Translations of Mathematical Monographs, 244, American Mathematical Society, 2014 | DOI

[34] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, 67, Springer, 1979, viii+241 pages (translated from the French by Marvin Jay Greenberg) | DOI | MR

[35] Sombra, Martín Minimums successifs des variétés toriques projectives, J. Reine Angew. Math., Volume 586 (2005), pp. 207-233 | DOI | Zbl

[36] Szpiro, Lucien; Ullmo, Emmanuel; Zhang, Shouwu Équirépartition des petits points, Invent. Math., Volume 127 (1997) no. 2, pp. 337-347 | DOI | MR | Zbl

[37] Ullmo, Emmanuel Positivité et discrétion des points algébriques des courbes, Ann. Math., Volume 147 (1998) no. 1, pp. 167-179 | DOI | MR | Zbl

[38] Yamaki, K. Survey on the geometric Bogomolov conjecture, Actes de la Conférence “Non-Archimedean Analytic Geometry: Theory and Practice” (Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres), Volume 2017/1, Presses Universitaires de Franche-Comté, 2017, pp. 137-193 | Numdam | MR | Zbl

[39] Yuan, Xinyi Big line bundles over arithmetic varieties, Invent. Math., Volume 173 (2008) no. 3, pp. 603-649 | DOI | MR | Zbl

[40] Zhang, Shouwu Positive line bundles on arithmetic varieties, J. Am. Math. Soc., Volume 8 (1995) no. 1, pp. 187-221 | DOI | MR | Zbl

[41] Zhang, Shouwu Small points and adelic metrics, J. Algebr. Geom., Volume 4 (1995) no. 2, pp. 281-300 | MR | Zbl

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

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