Uniqueness of the minimizer of the normalized volume function

Chenyang Xu, Ziquan Zhuang

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

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 in the local setting, 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.

Original languageEnglish (US)
Pages (from-to)149-176
Number of pages28
JournalCambridge Journal of Mathematics
Volume9
Issue number1
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • General Mathematics

Keywords

  • K-stability
  • Normalized volume
  • singularity

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