Critical Fubini-Study metrics over non-archimedean fields

TL;DR

This paper explores the properties of Fubini-Study metrics over non-archimedean fields, linking heights of projective varieties to Chow forms and characterizing metrics that minimize height via Monge-Ampère polytopes.

Contribution

It introduces a non-Archimedean Kempf-Ness criterion and characterizes height-minimizing Fubini-Study metrics using Monge-Ampère polytopes associated with polymatroids.

Findings

Height of a variety relates to its Chow form's naive height.

Characterization of height-minimizing metrics via Monge-Ampère polytopes.

Construction of polytopes using Bergman functionals or residual actions.

Abstract

Over a non-archimedean local place, the height of a projective variety with respect to a very ample line bundle equipped with a Fubini-Study metric is related to the naive height of its Chow form. Using a non-Archimedean Kempf-Ness criteria, we characterize Fubini-Study metrics that minimize the height under the special linear action in terms of their Monge-Amp\`ere polytopes. This polytope can be constructed either as its non-Archimedean Bergman functional or as the weight polytope for the residual action on its Chow form; it is associated with a polymatroid.