Global pluripotential theory for adelic line bundles

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.