Variation of canonical heights of subvarieties for polarized endomorphisms
Thomas Gauthier1 ; Gabriel Vigny2
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
Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1123-1135.

When an endomorphism f:X→X of a projective variety which is polarized by an ample line bundle L, i.e. such that f*L≃L⊗d with d≥2, is defined over a number field, Call and Silverman defined a canonical height h^f for f. In a family (𝒳,f,ℒ) parametrized by a curve S together with a section P:S→𝒳, they show that h^ft(P(t))/h(t) converges to the height h^fη(Pη) on the generic fiber.

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

Pour un endomorphisme f:X→X d’une variété projective qui est polarisé par un fibré en droites ample L, i.e. tel que f*L≃L⊗d avec d≥2, et qui est défini sur un corps de nombres, Call et Silverman ont défini une fonction hauteur canonique h^f. Dans une famille (𝒳,f,ℒ) paramétrée par une courbe S munie d’une section P:S→𝒳, ils prouvent également que h^ft(P(t))/h(t) converge vers la hauteur canonique h^fη(Pη) sur la fibre générique.

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

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DOI : 10.5802/jtnb.1310
Classification : 37P45, 14G40, 37F46
Mots-clés : Canonical height, families of polarized endomorphisms, bifurcations

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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Thomas Gauthier; Gabriel Vigny. Variation of canonical heights of subvarieties for polarized endomorphisms. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1123-1135. doi : 10.5802/jtnb.1310. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1310/

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