Variation of canonical heights of subvarieties for polarized endomorphisms

Gauthier, Thomas; Vigny, Gabriel

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

When an endomorphism $f:X\to X$ of a projective variety which is polarized by an ample line bundle $L$, i.e. such that $f^{*}L\simeq L^{\otimes d}$ with $d\ge 2$, is defined over a number field, Call and Silverman defined a canonical height ${\widehat{h}}_f$ for $f$. In a family $(𝒳,f,ℒ)$ parametrized by a curve $S$ together with a section $P:S\to 𝒳$, they show that ${\widehat{h}}_{f_t}\left(P\left(t\right)\right)/h\left(t\right)$ converges to the height ${\widehat{h}}_{f_{\eta }}\left(P_{\eta }\right)$ 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 $f^{*}L\simeq L^{\otimes d}$ avec $d\ge 2$, et qui est défini sur un corps de nombres, Call et Silverman ont défini une fonction hauteur canonique ${\widehat{h}}_f$. Dans une famille $(𝒳,f,ℒ)$ paramétrée par une courbe $S$ munie d’une section $P:S\to 𝒳$, ils prouvent également que ${\widehat{h}}_{f_t}\left(P\left(t\right)\right)/h\left(t\right)$ converge vers la hauteur canonique ${\widehat{h}}_{f_{\eta }}\left(P_{\eta }\right)$ 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.

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

[1] Berteloot, François; Bianchi, Fabrizio; Dupont, Christophe Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of ${\mathbb{P}}^k$, Ann. Sci. Éc. Norm. Supér. (4), Volume 51 (2018) no. 1, pp. 215-262 | DOI | MR | Zbl

[2] Boucksom, Sébastien; Eriksson, Dennis Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, Adv. Math., Volume 378 (2021), 107501, 125 pages | MR | Zbl

[3] Call, Gregory S.; Silverman, Joseph H. Canonical heights on varieties with morphisms, Compos. Math., Volume 89 (1993) no. 2, pp. 163-205 | MR | Numdam | Zbl

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

[5] Deligne, Pierre Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) (Contemporary Mathematics), Volume 67, American Mathematical Society, 1985, pp. 93-177 | DOI | MR

[6] DeMarco, Laura G. Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett., Volume 8 (2001) no. 1-2, pp. 57-66 | DOI | MR | Zbl

[7] Favre, Charles; Gauthier, Thomas Continuity of the Green function in meromorphic families of polynomials, Algebra Number Theory, Volume 12 (2018) no. 6, pp. 1471-1487 | DOI | MR | Zbl

[8] Gauthier, Thomas; Vigny, Gabriel The Geometric Dynamical Northcott and Bogomolov Properties (2019) (to appear in Ann. Sci. Éc. Norm. Supér.) | arXiv

[9] Ingram, Patrick Variation of the canonical height for a family of polynomials, J. Reine Angew. Math., Volume 685 (2013), pp. 73-97 | MR | Zbl

[10] Ingram, Patrick Variation of the canonical height in a family of polarized dynamical systems (2021) | arXiv

[11] Ingram, Patrick Explicit canonical heights for divisors relative to endomorphisms of ${\mathbb{P}}^n$, Matematica, Volume 3 (2024) no. 4, pp. 1456-1485 | Zbl | DOI | MR

[12] Lang, Serge Fundamentals of Diophantine geometry, Springer, 1983, xviii+370 pages | DOI | MR

[13] Lyubich, Mikhail Yu. Investigation of the stability of the dynamics of rational functions, Teor. Funkts., Funkts. Anal. Prilozh., Volume 42 (1984), pp. 72-91 translated in Sel. Math. Sov. 9 (1990), no. 1, p. 69–90 | MR | Zbl

[14] Mañé, Ricardo; Sad, Paolo; Sullivan, Denis On the dynamics of rational maps, Ann. Sci. Éc. Norm. Supér. (4), Volume 16 (1983) no. 2, pp. 193-217 | Numdam | DOI | MR | Zbl

[15] Raynaud, Michel; Gruson, Laurent Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | MR | Zbl

[16] Tate, John Variation of the canonical height of a point depending on a parameter, Am. J. Math., Volume 105 (1983) no. 1, pp. 287-294 | DOI | MR | Zbl

[17] Yuan, Xinyi; Zhang, Shou-Wu Adelic line bundles over quasi-projective varieties (2021) | arXiv

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

[19] Zhang, Shou-Wu Distributions in algebraic dynamics, Surveys in differential geometry. Vol. X (Surveys in Differential Geometry), Volume 10, International Press, 2006, pp. 381-430 | DOI | Zbl