Une généralisation du théorème de Liouville effectif pour les variétés projectives

Abstract

We prove an effective analogue of Liouville’s theorem for closed points on an arbitrary projective variety defined over a number field. Our result can be interpreted as an effective version of a recent theorem proved by McKinnon and Roth. A central part of our work here is dedicated to giving an effective proof of a particular case of a powerful theorem in diophantine geometry proved by Faltings and Wüstholz. This result, combined with new explicit comparisons between evaluation of sections of a line bundle and a given distance function, leads to the expected theorem. We also deal with another approach, showing how to make the arguments of McKinnon and Roth effective. These two points of view lead to distinct versions of our main result, giving different upper bounds for the height of points satisfying a Liouville-type inequality.