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Good height functions on quasi-projective varieties: equidistribution and applications in dynamics
[Bonnes fonctions hauteurs sur des variétés quasi-projectives : équidistribution et applications en dynamique]
Thomas Gauthier1
1Laboratoire de Mathématiques d’Orsay, Bâtiment 307, rue Michel Magat, Faculté des Sciences d’Orsay, Université Paris-Saclay F-91405 Orsay Cedex France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 1, pp. 57-94

In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $\mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions.

Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of Kühne and Yuan and Zhang, we deduce the equidistribution of generic sequence of preperiodic parameters for families of polarized endomorphisms with marked points.

Dans cet article, nous définissons une notion de bonne fonction hauteur sur une variété quasi-projective $V$ définie sur un corps de nombres et nous prouvons un théorème d’équidistribution des petits points pour de telles fonctions hauteurs. Ces bonnes fonctions hauteurs sont définies comme des limites de fonctions hauteurs associées à des suites de $\mathbb{Q}$-fibrés en droites munis de métrisations adéliques semi-positives sur des modèles projectifs de $V$ satisfaisant des hypothèses assez générales.

En nous appuyant sur un récent travail de l’auteur et Vigny, ainsi que sur des estimées classiques de Call et Silverman, et en nous inspirant de travaux récents de Kühne et de Yuan et Zhang, nous en déduisons un résultat d’équidistribution pour les suites génériques de paramètres prépériodiques pour des familles d’endomorphismes polarisés munis de points marqués.

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DOI : 10.5802/afst.1841

Thomas Gauthier  1

1 Laboratoire de Mathématiques d’Orsay, Bâtiment 307, rue Michel Magat, Faculté des Sciences d’Orsay, Université Paris-Saclay F-91405 Orsay Cedex France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Thomas Gauthier. Good height functions on quasi-projective varieties: equidistribution and applications in dynamics. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 1, pp. 57-94. doi: 10.5802/afst.1841

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