Uniqueness of the minimizer of the normalized volume function

Abstract

We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a valuation is a minimizer if and only if it is K-semistable, and that K-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of klt singularities and an effective bound of the local fundamental group of a klt singularity.

arXiv comment20 pages

References in corpus (4)

• Finiteness of algebraic fundamental groups
• Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below
• Existence of valuations with smallest normalized volume
• The local fundamental group of a Kawamata log terminal singularity is finite

Cited by in corpus (8)

• Optimal destabilization of K-unstable Fano varieties via stability thresholds
• The existence of the Kähler-Ricci soliton degeneration
• The Jordan property for local fundamental groups
• On K-stability of some del Pezzo surfaces of Fano index 2
• K-stability of cubic fourfolds
• A Note on Equivariant K-stability
• Volume and Monge-Ampère energy on polarized affine varieties
• On termination of flips and fundamental groups