Classically psh and pluriharmonic functions on Berkovich spaces

TL;DR

This paper extends the theory of subharmonic, plurisubharmonic, and pluriharmonic functions to possibly singular Berkovich spaces over non-archimedean fields, establishing foundational properties and finiteness results.

Contribution

It generalizes classical psh function theory to singular Berkovich spaces and proves the finite dimensionality of pluriharmonic functions on such spaces.

Findings

Extended subharmonic function theory to singular curves.

Established maximum principles for psh functions.

Proved finiteness of pluriharmonic function space.

Abstract

First we extend the theory of subharmonic functions on smooth strictly $k$-analytic curves from Thuillier's thesis to the case of possibly singular analytic curves over a non-archimedean field. Classically psh functions are then defined as in complex geometry by using pullbacks to analytic curves (and requiring compatibility with base change). We give various properties of classically psh functions including a local and a global maximum principle. As a consequence, we show that the space of pluriharmonic functions on a quasi-compact Berkovich space is finite dimensional. As a technical tool, we use that a connected Berkovich space is connected by analytic curves.