Global pluripotential theory for adelic line bundles

TL;DR

This paper connects adelic line bundles with pluripotential theory on Berkovich spaces, establishing an equivalence and applying it to Monge--Ampère measures and subvarieties in arithmetic geometry.

Contribution

It introduces a new equivalence between adelic line bundles and line bundles with psh metrics on Berkovich spaces, extending pluripotential theory to quasi-projective arithmetic varieties.

Findings

Established an equivalence between adelic line bundles and line bundles with psh metrics on Berkovich spaces.

Generalized Monge--Ampère measure constructions to quasi-projective arithmetic varieties.

Provided a new description of non-degenerate subvarieties via Monge--Ampère measures.

Abstract

In this work, we relate recent work of Yuan--Zhang and Song on adelic line bundles over quasi-projective arithmetic varieties to recent advances in pluripotential theory on global Berkovich spaces from Pille-Schneider. In particular, we establish an equivalence between subcategories of adelic line bundles on quasi-projective varieties and line bundles on their Berkovich analytifications equipped with a continuous plurisubharmonic metric. We also provide several applications of this equivalence. For example, we generalize a construction of Pille-Schneider concerning families of Monge--Amp\`ere measures on analytifications of projective arithmetic varieties to the quasi-projective setting. With this construction, we offer a new description of non-degenerate subvarieties which involves Monge--Amp\`ere measures over trivially valued fields. Finally, we define a Monge--Amp\`ere measure on the analytification of a quasi-projective arithmetic variety.