Variation of canonical heights of subvarieties for polarized endomorphisms

Thomas Gauthier; Gabriel Vigny

doi:10.5802/jtnb.1310

¹LMO, Université Paris-Saclay Bâtiment 307, rue Michel Magat 91405 Orsay Cedex, France
²LAMFA, Université de Picardie Jules Verne 33 rue Saint-Leu 80039 Amiens Cedex 1, France

Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 3, pp. 1123-1135

Abstract (Original language)
Résumé

When an endomorphism $f : X \to X$ of a projective variety which is polarized by an ample line bundle $L$ , i.e. such that $f^{*} L ≃ L^{\otimes d}$ with $d \geq 2$ , is defined over a number field, Call and Silverman defined a canonical height ${\hat{h}}_{f}$ for $f$ . In a family $(𝒳, f, ℒ)$ parametrized by a curve $S$ together with a section $P : S \to 𝒳$ , they show that ${\hat{h}}_{f_{t}} (P (t)) / h (t)$ converges to the height ${\hat{h}}_{f_{η}} (P_{η})$ on the generic fiber.

In the present paper, we prove the equivalent statement when studying the variation of canonical heights of subvarieties $Y_{t}$ varying in a family $𝒴$ of any relative dimension.

Pour un endomorphisme $f : X \to X$ d’une variété projective qui est polarisé par un fibré en droites ample $L$ , i.e. tel que $f^{*} L ≃ L^{\otimes d}$ avec $d \geq 2$ , et qui est défini sur un corps de nombres, Call et Silverman ont défini une fonction hauteur canonique ${\hat{h}}_{f}$ . Dans une famille $(𝒳, f, ℒ)$ paramétrée par une courbe $S$ munie d’une section $P : S \to 𝒳$ , ils prouvent également que ${\hat{h}}_{f_{t}} (P (t)) / h (t)$ converge vers la hauteur canonique ${\hat{h}}_{f_{η}} (P_{η})$ sur la fibre générique.

Dans cet article, nous étudions les variations en famille de la hauteur canonique de sous-variétés $Y_{t}$ et nous démontrons un énoncé équivalent en toute dimension relative.

Received: 2023-07-20
Revised: 2023-12-11
Accepted: 2023-12-15
Published online: 2025-03-31

DOI: 10.5802/jtnb.1310

Classification: 37P45, 14G40, 37F46
Keywords: Canonical height, families of polarized endomorphisms, bifurcations

Author's affiliations:

Thomas Gauthier ¹ ; Gabriel Vigny ²

¹ LMO, Université Paris-Saclay Bâtiment 307, rue Michel Magat 91405 Orsay Cedex, France
² LAMFA, Université de Picardie Jules Verne 33 rue Saint-Leu 80039 Amiens Cedex 1, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Thomas Gauthier; Gabriel Vigny. Variation of canonical heights of subvarieties for polarized endomorphisms. Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 3, pp. 1123-1135. doi: 10.5802/jtnb.1310

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