Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents

Abstract

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a metrized vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety—provided this coherent sheaf is a subquotient of a metrized vector bundle—and, using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of the so constructed arithmetic Hilbert-Samuel function.