Transcendental Liouville Inequalities on Projective Varieties

Abstract

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 a hermitian line bundle over $X$ is 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 Chudnovsky 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.