M\'etriques de sous-quotient et th\'eor\`eme de Hilbert-Samuel arithm\'etique pour les faisceaux coh\'erents

TL;DR

This paper extends sections of coherent subquotients of hermitian vector bundles on complex spaces without smoothness assumptions and applies this to define and analyze an arithmetic Hilbert-Samuel function for coherent sheaves on arithmetic varieties.

Contribution

It proves a new extension theorem for sections of coherent subquotients without smoothness constraints and introduces an arithmetic Hilbert-Samuel function for such sheaves, deriving its leading term.

Findings

Extension theorem for coherent subquotients without smoothness assumptions

Construction of an arithmetic Hilbert-Samuel function for coherent sheaves

Determination of the leading term of the arithmetic Hilbert-Samuel function

Abstract

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian 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 hermitian 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.