Uniqueness of the minimizer of the normalized volume function
Chenyang Xu, Ziquan Zhuang
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.
Topics & Concepts
MathematicsUniquenessGravitational singularityConjectureSingularityValuation (finance)Pure mathematicsFunction (biology)Upper and lower boundsStability (learning theory)Degree (music)Mathematical analysisApplied mathematicsComputer scienceAcousticsEconomicsFinancePhysicsEvolutionary biologyMachine learningBiologyGeometry and complex manifoldsAdvanced Topology and Set TheoryAlgebraic Geometry and Number Theory