Ample canonical heights for endomorphisms on projective varieties
Abstract
We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell--Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety.
Cite
@article{arxiv.1710.05278,
title = {Ample canonical heights for endomorphisms on projective varieties},
author = {Takahiro Shibata},
journal= {arXiv preprint arXiv:1710.05278},
year = {2018}
}
Comments
41 pages. The previous version has a serious mistake on the proof of the main conjecture for simple abelian varieties, but the present version gives a renewed proof that works for arbitrary abelian varieties