Good height functions on quasi-projective varieties: equidistribution and applications in dynamics

TL;DR

This paper introduces a new class of height functions on quasi-projective varieties over number fields, proves an equidistribution theorem for small points, and applies these results to dynamics, specifically preperiodic points in families of endomorphisms.

Contribution

It defines good height functions as limits of heights from semi-positive adelic metrizations and establishes their equidistribution properties, extending previous work in arithmetic dynamics.

Findings

Proved equidistribution of small points for good height functions.

Applied results to preperiodic points in families of polarized endomorphisms.

Extended classical estimates to broader classes of height functions.

Abstract

In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $\mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of K\"uhne and Yuan and Zhang, we deduce the equidistribution of generic sequence of preperiodic parameters for families of polarized endomorphisms with marked points.