Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

TL;DR

This paper proves that for certain self-maps on projective varieties, the growth rate of heights associated with a subscheme is negligible compared to ample heights along orbits, under specific dynamical and geometric conditions.

Contribution

It establishes a limit relation between subscheme heights and ample heights for orbits of self-rational maps, extending previous dynamical height results to more general settings.

Findings

The ratio of heights tends to zero along orbits under given conditions.

The result applies to morphisms and certain regular embeddings.

It links dynamical degrees and arithmetic degrees to height growth.

Abstract

Consider a dominant rational self-map $f$ on a smooth projective variety $X$ defined over $\overline{\mathbb{Q}}$. We prove that \begin{align} \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x)) } = 0, \end{align} where $h_{Y}$ is a height associated with a closed subscheme $Y \subset X$ of codimension $c$, $h_{H}$ is any ample height on $X$, and $x \in X(\overline{\mathbb{Q}})$ is a point with well-defined orbit, under the following assumptions: (1) either $f$ is a morphism, or $Y$ is pure dimensional, regularly embedded in $X$, and contained in the locus over which all iterates of $f$ are finite; (2) the orbit of $x$ is generic; (3) $d_{c}(f)^{1/c} < \alpha_{f}(x)$, where $d_{c}(f)$ is the $c$-th dynamical degree of $f$ and $ \alpha_{f}(x)$ is the arithmetic degree of $x$.