The semicontinuity problem of normalized volume of singularities
The semicontinuity 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 QGorenstein 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 Ksemistability developed by Li, Xu and the speaker, we show that Ksemistability is a very generic property in a QGorenstein flat family of QFano varieties.