Metrization of differential pluriforms on Berkovich analytic spaces

TL;DR

This paper introduces a seminorm called Kahler seminorm on sheaves of differential pluriforms in Berkovich spaces, extending known results about pluricanonical forms and their maximality loci in non-Archimedean geometry.

Contribution

It defines a new seminorm on sheaves of pluriforms and extends the understanding of maximality loci of pluricanonical forms to more general Berkovich spaces.

Findings

Maximality locus of pluricanonical forms is a PL subspace contained in the skeleton.

Kahler seminorm coincides with the weight norm in certain cases.

Extension of Mustata and Nicaise's results to quasi-smooth Berkovich spaces.

Abstract

We introduce a general notion of a seminorm on sheaves of rings or modules and provide each sheaf of relative differential pluriforms on a Berkovich k-analytic space with a natural seminorm, called Kahler seminorm. If the residue field is of characteristic zero and X is a quasi-smooth k-analytic space, then we show that the maximality locus of any global pluricanonical form is a PL subspace of X contained in the skeleton of any semistable formal model of X. This extends a result of Mustata and Nicaise, because the Kahler seminorm on pluricanonical forms coincides with the weight norm defined by Mustata and Nicaise when k is discretely valued and of residue characteristic zero.