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Discreteness of local volumes

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EMGW04 - K-stability and moment maps

(Joint with Ziquan Zhuang) The concept of local volume of a klt singularity, which is the infimum of normalized volumes of all valuations centered at the singularity, plays a central role in the local stability theory. The Stable Degeneration Theorem says any klt singularity admits a volume preserved degeneration to a K-semistable log Fano cone singularity. In this talk, I will show that all K-semistable log Fano cone singularities with volume bounded from below are bounded. As a consequence, this implies that the set of normalized volumes of all klt singularities has 0 as its only accumulation point.  

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