On the boundedness of singularities via normalized volume

Joaquín Moraga, Hendrik Süß

Published 2022-05-24Version 1

In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on hypersurface singularities and singularities with torus action of complexity one. Given a positive integer n and a real value v>0, we prove that the class of n-dimensional K-semistable hypersurface singularities with normalized volume at least v forms a bounded family. Analogously, we prove that the class of n-dimensional K-semistable singularities with torus action of complexity one and with normalized volume at least v forms a bounded family. 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 complexity one torus action. We exhibit a 3-dimensional example which shows that this last statement is optimal.