On the boundedness of singularities via normalized volume

TL;DR

This paper investigates the boundedness of singularities using normalized volume, proving boundedness results for certain classes like K-semistable threefolds and hypersurface singularities, and exploring the role of torus actions.

Contribution

It establishes boundedness of classes of singularities with normalized volume constraints, extending results to higher dimensions and specific symmetry conditions.

Findings

K-semistable threefold singularities with volume ≥ v are bounded.

Boundedness results extend to n-dimensional complexity-1 and hypersurface singularities.

A 3D example shows the optimality of the boundedness statement.

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.