2019/12/17 by Gauthier, Thomas, Vigny, Gabriel · 1 citation
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1912.07907
We establish the dynamical Northcott property for polarized endomorphisms of a projective variety over a function field 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 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.