Majorations explicites de fonctions de Hilbert-Samuel g\'eom\'etrique et arithm\'etique

TL;DR

This paper develops explicit upper bounds for the Hilbert-Samuel functions in both geometric and arithmetic contexts using Arakelov geometry's R-filtration approach, advancing understanding of line bundles on projective varieties.

Contribution

It introduces a method to derive explicit bounds for Hilbert-Samuel functions in geometric and arithmetic settings via R-filtration in Arakelov geometry.

Findings

Established explicit upper bounds for geometric Hilbert-Samuel functions.

Derived explicit bounds for arithmetic Hilbert-Samuel functions.

Applied R-filtration approach to line bundles on projective varieties.

Abstract

By using the $\mathbb R$-filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic projective varieties.