Transcendental Liouville inequalities on projective varieties

Let be an algebraic point of a projective variety defined over a number field. Liouville inequality tells us that the norm at of a non-vanishing integral global section of a hermitian line bundle over is zero or it cannot be too small with respect to the 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.