The normalized volume of a singularity is lower semicontinuous

Abstract

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.

28 pages. Comments are very welcome. v2: minor changes. To appear in J. Eur. Math. Soc. (JEMS). Remark: this preprint improves main results of the preprint (arXiv:1711.06962) by the second author

References in corpus (3)

• Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties
• Finiteness of étale fundamental groups by reduction modulo $p$
• Seifert $G_m$-bundles

Cited by in corpus (12)

• A non-Archimedean approach to K-stability
• Optimal destabilization of K-unstable Fano varieties via stability thresholds
• Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties
• Birational superrigidity and K-stability of singular Fano complete intersections
• Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)
• Wall crossing for K-moduli spaces of plane curves
• Equivariant K-stability under finite group action
• K-stability of cubic fourfolds
• A Minimizing Valuation is Quasi-monomial
• On the sharpness of Tian's criterion for K-stability
• On boundedness of singularities and minimal log discrepancies of Kollár components, II
• Stable degeneration of families of klt singularities with constant local volume