On the semi-continuity problem of normalized volumes of singularities

2017/11/19 by Yuchen Liu, Liu, Yuchen · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1711.06962

openalex publication_date 2017/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that in any ℚ-Gorenstein flat family of klt singularities, normalized volumes can only jump down at countably many subvarieties. 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, Xu and the author, we show that K-semistability is a very generic or empty condition in any ℚ-Gorenstein flat family of log Fano pairs.

Citations

• Seifert G_m-bundles
• Finiteness of étale fundamental groups by reduction modulo p

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• Interaction Between Singularity Theory and the Minimal Model Program

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