The semi-continuity problem of normalized volume of singularities

Yuchen Liu , Yale University
Fine Hall 314

Motivated by work in differential geometry, Chi Li introduced the normalized volume of a klt singularity as the minimum normalized volume of all valuations centered at the singularity. This invariant carries some interesting geometric/topological information of the singularity. In this talk, we show that in a Q-Gorenstein flat family of klt singularities, normalized volumes only jump down at(possibly) countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using the valuative characterization of K-semistability developed by Li, Xu and the speaker, we show that K-semistability is a very generic property in a Q-Gorenstein flat family of Q-Fano varieties.