Cambridge Journal of Mathematics

Volume 9 (2021)

Number 1

Uniqueness of the minimizer of the normalized volume function

Pages: 149 – 176

DOI: https://dx.doi.org/10.4310/CJM.2021.v9.n1.a2

Authors

Chenyang Xu (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.; Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.; and Beijing International Center for Mathematical Research, Beijing, China)

Ziquan Zhuang (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Abstract

We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a $\mathrm{klt}$ 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 $\mathrm{K}$-semistable, and that $\mathrm{K}$-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of $\mathrm{klt}$ singularities and an effective bound of the local fundamental group of a $\mathrm{klt}$ singularity.

Keywords

normalized volume, $\mathrm{K}$-stability, singularity

2010 Mathematics Subject Classification

Primary 14B05. Secondary 13A18.

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C. Xu was partially supported by the NSF (DMS-1901849 and DMS-1952531).

Received 21 May 2020

Published 27 October 2021