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Journal of the European Mathematical Society


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Volume 23, Issue 4, 2021, pp. 1225–1256
DOI: 10.4171/JEMS/1032

Published online: 2020-12-22

The normalized volume of a singularity is lower semicontinuous

Harold Blum[1] and Yuchen Liu[2]

(1) University of Michigan, Ann Arbor, USA
(2) Yale University, New Haven, USA

We show that in any $\mathbb Q$-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Liu, and Xu, we show that K-semistability is a very generic or empty condition in any $\mathbb Q$-Gorenstein flat family of log Fano pairs.

Keywords: Singularities, normalized volume, K-stability

Blum Harold, Liu Yuchen: The normalized volume of a singularity is lower semicontinuous. J. Eur. Math. Soc. 23 (2021), 1225-1256. doi: 10.4171/JEMS/1032