DOI: 10.1112/jlms.70738 ISSN: 0024-6107

Optimal bounds for local volumes of threefold singularities

Yuchen Liu
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Abstract

We establish an optimal upper bound for local volumes of Gorenstein canonical non‐hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least 9 is either a hypersurface singularity or a quotient singularity. As applications, we obtain new restrictions on the singularities of members in K‐moduli spaces of Fano threefolds, and we establish a sharp inequality between local volumes and minimal log discrepancies for threefold singularities.

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