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arXiv · 2005.08303

Uniqueness of the minimizer of the normalized volume function

Chenyang Xu·Ziquan Zhuang
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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.

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Chenyang Xu, Ziquan Zhuang. 2020-05-17. Uniqueness of the minimizer of the normalized volume function. https://arxiv.org/abs/2005.08303

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