We show that in any -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 -Gorenstein flat family of log Fano pairs.
Cite this article
Harold Blum, Yuchen Liu, The normalized volume of a singularity is lower semicontinuous. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1225–1256DOI 10.4171/JEMS/1032