On variation of dynamical canonical heights, and Intersection numbers

TL;DR

This paper investigates how canonical heights vary in families of varieties with multiple maps, providing explicit formulas for their dependence on parameters and linking local heights to intersection numbers under certain models.

Contribution

It extends Kawaguchi's work on canonical heights to families with multiple maps and relates local heights to intersection theory using weak Néron models.

Findings

Explicit dependence of canonical heights on parameters in families.

Extension of Néron and Silverman's results to systems with multiple maps.

Representation of local heights as intersection numbers under certain models.

Abstract

We study families of varieties endowed with polarized canonical eigensystems of several maps, inducing canonical heights on the dominating variety as well as on the "good" fibers of the family. We show explicitely the dependence on the parameter for global and local canonical heights defined by Kawaguchi when the fibers change, extending previous works of J. Silverman and others. Finally, fixing an absolute value $v \in K$ and a variety $V/K$, we descript the Kawaguchi`s canonical local height $\hat{\lambda}_{V,E,\mathcal{Q},}(.,v)$ as an intersection number, provided that the polarized system $(V,\mathcal{Q})$ has a certain weak N\'{e}ron model over Spec$(\mathcal{O}_v)$ to be defined and under some conditions depending on the special fiber. With this we extend N\'{e}ron's work strengthening Silverman's results, which were for systems having only one map.