Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

TL;DR

This paper establishes that the space of finite-energy non-Archimedean metrics on a polarized variety forms a geodesic metric space with continuous geodesics, extending the understanding of metric geometry in non-Archimedean settings.

Contribution

It proves the geodesic structure of finite-energy metrics and the continuity of maximal segments between continuous metrics in non-Archimedean geometry.

Findings

Finite-energy metrics form a geodesic metric space.

Maximal psh segments are continuous between continuous metrics.

The results extend metric geometry concepts to non-Archimedean contexts.

Abstract

Given a polarized projective variety (X,L) over any non-Archimedean field, assuming continuity of envelopes, we show that the space of finite-energy metrics on L is a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous.