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arXiv · 2205.12326

On the boundedness of singularities via normalized volume

Yuchen Liu·Joaquín Moraga·Hendrik Süß
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Abstract

In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value v>0, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-1 and n-dimensional hypersurface singularities for arbitary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with a complexity-1 torus action. We exhibit a 3-dimensional example which shows that this last statement is optimal.

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Yuchen Liu, Joaquín Moraga, Hendrik Süß. 2022-05-24. On the boundedness of singularities via normalized volume. https://arxiv.org/abs/2205.12326

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