The Geometric Dynamical Northcott and Bogomolov Properties

Abstract

We establish the dynamical Northcott property for polarized endomorphisms of a projective variety over a function field $\mathbf{K}$ of characteristic zero, and we relate this property to the notion of stability in complex dynamics. This extends previous results of Benedetto, Baker and DeMarco in dimension $1$, and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamics arguments and does not rely on the previous ones. We first show that, when $\mathbf{K}$ is the field of rational functions of a normal complex projective variety, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the geometric dynamical Northcott property characterizing points of height zero in this setting, using a similarity argument. Moving from points to subvarieties, we propose, for polarized endomorphisms, a dynamical version of the geometric Bogomolov conjecture, recently proved by Cantat, Gao, Habegger, and Xie in the original setting of abelian varieties.

accepted version

References in corpus (7)

• Questions on self maps of algebraic varieties
• The Bogomolov conjecture for totally degenerate abelian varieties
• Dynamical Degree and Arithmetic Degree of Endomorphisms on Product Varieties
• Lyapunov exponents and bifurcation current for polynomial-like maps
• Equidistribution of Dynamically Small Subvarieties over the Function Field of a Curve
• The geometric Bogomolov conjecture
• Higher arithmetic degrees of dominant rational self-maps

Cited by in corpus (7)

• Equidistribution in Families of Abelian Varieties and Uniformity
• Adelic line bundles on quasi-projective varieties
• A Tits alternative for rational functions
• Fixed points of Koch's maps
• Variation of the canonical height in a family of polarized dynamical systems
• Geometric methods in holomorphic dynamics
• Periodic points of post-critically algebraic endomorphisms