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arXiv · 2512.05429

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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Yuchen Liu. 2025-12-05. Optimal bounds for local volumes of threefold singularities. https://arxiv.org/abs/2512.05429

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