Gap for geometric canonical height functions

TL;DR

This paper establishes a gap around zero for canonical height functions in complex function fields and links zero-height points to birational isotriviality, extending Northcott property results to higher dimensions.

Contribution

It proves a zero-gap for canonical heights and characterizes when zero-height points are Zariski dense, connecting to birational isotriviality of endomorphisms.

Findings

Existence of a gap around zero for canonical heights.

Zero-height points are Zariski dense iff the endomorphism is birationally isotrivial.

A geometric Northcott property for projective plane height functions.

Abstract

We prove the existence of a gap around zero for canonical height functions associated to endomorphisms of projective spaces defined over complex function fields. We also prove that if the rational points of height zero are Zariski dense, then the endomorphism is birationally isotrivial. As a corollary, by a result of S. Cantat and J. Xie, we have a geometric Northcott property on projective plane in the same spirit of results of R. Benedetto, M. Baker and L. Demarco on the projective line.