ON THE BOUNDEDNESS OF SINGULARITIES VIA NORMALIZED VOLUME

Source abstract

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 $1$ 1 , threefold singularities, and hypersurface singularities. Given a real value v > 0 $v>0$ v greater than 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 $1$ 1 and n -dimensional hypersurface singularities for arbitrary 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 $1$ 1 torus action. We exhibit a 3 $3$ 3 -dimensional example which shows that this last statement is optimal.