Optimal bounds for local volumes of threefold singularities

TL;DR

This paper establishes an optimal upper bound for local volumes of Gorenstein canonical threefold singularities, revealing that high local volume singularities are either hypersurface or quotient types, with implications for Fano threefold moduli.

Contribution

It provides the first sharp bounds linking local volumes and singularity types in threefolds, and applies these bounds to K-moduli and minimal log discrepancies.

Findings

High local volume threefold singularities are hypersurface or quotient singularities.

Established a sharp inequality between local volumes and minimal log discrepancies.

Derived new restrictions on singularities in K-moduli spaces of Fano threefolds.

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.