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We establish the Geometric Dynamical Northcott Property for polarizedendomorphisms of a projective normal variety over a function field $\mathbf{K}$of characteristic zero. This extends previous results of Benedetto, Baker andDeMarco in dimension $1$, and of Chatzidakis-Hrushovski in higher dimension.Our proof uses complex dynamics arguments and does not rely on the previousone. We first show that when $\mathbf{K}$ is the field of rational functions ofa smooth complex projective variety, the canonical height of a subvariety isthe mass of the appropriate bifurcation current and that a marked point isstable if and only if its canonical height is zero. We then establish theGeometric Dynamical Northcott Property using a similarity argument.  Moving from points to subvarieties, we propose, for polarized endomorphisms,a dynamical version of the Geometric Bogomolov Conjecture, recently proved byCantat, Gao, Habegger and Xie. We establish several cases of this conjecturenotably non-isotrivial polynomial skew-product with an isotrivial firstcoordinate.

The Geometric Dynamical Northcott and Bogomolov Properties