Transcendental Liouville inequalities on projective varieties

TL;DR

This paper extends Liouville inequalities to transcendental points on projective varieties, showing most such points satisfy strong inequalities and exploring implications for rational point distribution.

Contribution

It introduces Liouville-type inequalities for transcendental points and relates these to conjectures and rational point counting on varieties.

Findings

Most transcendental points satisfy Liouville inequalities.

Connections established with Chudnowsky's conjecture.

Applications to counting rational points of bounded height.

Abstract

Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non vanishing integral global section of an hermitian line bundle over $X$ is either zero or it cannot be too small with respect to the $\sup$ norm of the section itself. We study inequalities similar to Liouville's for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnowsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.