The normalized volume of a singularity is lower semicontinuous
Algebraic Geometry
2021-07-14 v2 Commutative Algebra
Differential Geometry
Abstract
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
@article{arxiv.1802.09658,
title = {The normalized volume of a singularity is lower semicontinuous},
author = {Harold Blum and Yuchen Liu},
journal= {arXiv preprint arXiv:1802.09658},
year = {2021}
}
Comments
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