Pluripotential-theoretic stability thresholds

Abstract

Given a compact polarized manifold (X, L), we introduce two new stability thresholds in terms of singularity types of global quasi-plurisubharmonic functions on X. We prove that in the Fano setting, the new invariants can effectively detect the K-stability of X. We study some functionals of geodesic rays in the space of Kähler potentials by means of the corresponding test curves. In particular, we introduce a new entropy functional of quasi-plurisubharmonic functions and relate the radial entropy functional to this new entropy functional.

Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics

“…With some efforts, we can reduce the problem to the case where φ has analytic singularities along some normal crossing (nc) Q-divisor on X and dd c φ is a Kähler current. In this case, the desired result follows from a result proved in [Xia20].…”

“…Lastly, let us mention that our generalization of Boucksom-Chen theorem has important consequences in Archimedean pluripotential theory as well. When applied to generalized deformation to the normal cone in the sense of [Xia20], it gives a number of interesting equidistribution results of the jumping numbers of multiplier ideal sheaves. As a detailed investigation would lead us too far away, we do not include these results in this paper.…”

“…Perturbing L by an ample Q-line bundle by Proposition 5.17, we may assume that θ ϕ is a Kähler current. Finally, by rescaling, we may assume that F is a divisor and L is a line bundle and L − F is ample by [Xia20,Lemma 2.4].…”