A transcendental approach to non-Archimedean metrics of pseudoeffective classes

Tamás Darvas; Mingchen Xia; Kewei Zhang

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arXiv · 2302.02541

A transcendental approach to non-Archimedean metrics of pseudoeffective classes

Tamás Darvas·Mingchen Xia·Kewei Zhang

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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.

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BibTeX RIS

Tamás Darvas, Mingchen Xia, Kewei Zhang. 2023-02-06. A transcendental approach to non-Archimedean metrics of pseudoeffective classes. https://arxiv.org/abs/2302.02541

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