Concave transforms of compactified S-metrized divisors

TL;DR

This paper introduces a method to associate concave transforms to compactified S-metrized divisors on quasi-projective varieties over adelic curves, and establishes a Hilbert-Samuel type formula for relatively nef cases.

Contribution

It presents a novel approach linking concave transforms with S-metrized divisors and provides a new formula generalizing Hilbert-Samuel theory in this context.

Findings

Established a Hilbert-Samuel type formula for relatively nef S-metrized divisors.

Defined a concave transform associated with compactified S-metrized divisors.

Extended classical geometric formulas to adelic and metrized divisor settings.

Abstract

We associate a concave transform to any compactified S-metrized divisor on a quasi-projective variety over an adelic curve. Then we show a Hilbert-Samuel type formula for relatively nef compactified S-metrized YZ-divisors.