Pluripotential-theoretic stability thresholds

TL;DR

This paper introduces new stability thresholds for polarized manifolds using singularity types of quasi-plurisubharmonic functions, linking them to K-stability in the Fano case and exploring related entropy functionals.

Contribution

It defines novel stability invariants based on singularity types and demonstrates their effectiveness in detecting K-stability for Fano manifolds.

Findings

New stability thresholds effectively detect K-stability in Fano manifolds.

Introduction of a new entropy functional for quasi-plurisubharmonic functions.

Relationship established between radial entropy and the new entropy functional.

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 K\"ahler 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.