JournalsjemsVol. 23, No. 4pp. 1225–1256

The normalized volume of a singularity is lower semicontinuous

  • Harold Blum

    University of Michigan, Ann Arbor, USA
  • Yuchen Liu

    Yale University, New Haven, USA
The normalized volume of a singularity is lower semicontinuous cover
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Abstract

We show that in any Q\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 Q\mathbb Q-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. 1225–1256

DOI 10.4171/JEMS/1032