A transcendental approach to non-Archimedean metrics of pseudoeffective classes

2023/02/06 by Tamás Darvas, Darvas, Tamás, Mingchen Xia +3 · 5 citations
Mathematics · #Geometry and complex manifolds #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2302.02541

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 in terms of flag configurations.

Cited by

• Relative pluripotential theory on compact Kähler manifolds
• Two Calabi-Yau theorems for degenerations of compact Kähler manifolds
• A YTD correspondence for constant scalar curvature metrics
• A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem
• Lines in the space of Kähler metrics

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