A transcendental approach to non-Archimedean metrics of pseudoeffective classes

TL;DR

This paper develops a new transcendental non-Archimedean metric framework for pseudoeffective classes on Kähler manifolds, extending existing correspondences and introducing flag configurations to analyze stability and energy functionals.

Contribution

It extends the Ross-Witt Nystr"om correspondence to the transcendental setting and introduces flag configurations for big classes, connecting them to stability and energy concepts.

Findings

Non-Archimedean finite energy metrics are approximable by flag configurations.

Very general radial Ding energies are continuous, even in the ample case.

Characterization of the delta invariant as Ding semistability threshold.

Abstract

We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact K\"ahler manifold. This is obtained via extending the Ross-Witt Nystr\"om correspondence to the relative case, and we point out that our construction agrees with that of Boucksom-Jonsson when the class is induced by a pseudoeffective $\mathbb Q$-line bundle. We introduce the notion of a flag configuration attached to a transcendental big class, recovering the notion of a test configuration in the ample case. We show that non-Archimedean finite energy metrics are approximable by flag configurations, and very general versions of the radial Ding energy are continuous, a novel result even in the ample case. As applications, we characterize the delta invariant as the Ding semistability threshold of flag configurations and filtrations and prove a YTD type existence theorem in terms of flag configurations.