Hilbert-Samuel formula and positivity over adelic curves

TL;DR

This paper extends the Hilbert-Samuel formula to adelic curves in Arakelov geometry, linking the asymptotic growth of line bundles to their intersection numbers and exploring positivity conditions.

Contribution

It introduces an arithmetic Hilbert-Samuel theorem for adelic line bundles and analyzes their positivity properties in Arakelov geometry.

Findings

Asymptotic behavior of metrized graded linear series described

Arithmetic intersection number linked to growth of line bundles

Positivity conditions of adelic line bundles studied

Abstract

We establish, in the setting of Arakelov geometry over adelic curves, an arithmetic Hilbert-Samuel theorem describing the asymptotic behaviour of the metrized graded linear series of an adelic line bundle in terms of its arithmetic intersection number. We then study positivity conditions of adelic line bundles.