The closures of test configurations and algebraic singularity types

Abstract

Given a Kähler manifold $X$ with an ample line bundle $L$, we consider the metric space of $L^1$ geodesic rays associated to the first Chern class $c_1(L)$. We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge--Ampère energy of a general $L^1$ ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations. Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge--Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from $L^1$ rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays.

arXiv commentv.2

References in corpus (15)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability
• Effectiveness of Demailly's strong openness conjecture and related problems
• The openness conjecture for plurisubharmonic functions
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact Kähler manifolds
• On the constant scalar curvature Kähler metrics, existence results
• On the constant scalar curvature Kähler metrics, general automorphism group
• A non-Archimedean approach to K-stability
• On the constant scalar curvature Kähler metrics, apriori estimates
• Geodesic rays and stability in the cscK problem
• Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)
• Mabuchi geometry of big cohomology classes with prescribed singularities
• Jumping numbers of analytic multiplier ideals (with an appendix by Sébastien Boucksom)
• Tian's properness conjectures: an introduction to Kahler geometry

Cited by in corpus (6)

• Geodesic rays and stability in the cscK problem
• Kähler-Einstein metrics with prescribed singularities on Fano manifolds
• Emergent complex geometry
• Valuative invariants with higher moments
• Deformations of Kähler manifolds to normal bundles and restricted volumes of big classes
• Plurisupported Currents on Compact Kähler Manifolds