A transcendental approach to non-Archimedean metrics of pseudoeffective classes

Abstract

We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact Kähler manifold. This is obtained via extending the Ross–Witt Nyström 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 for Kähler–Einstein metrics in terms of flag configurations.