On the boundedness of singularities via normalized volume
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.
Cite
@article{arxiv.2205.12326,
title = {On the boundedness of singularities via normalized volume},
author = {Yuchen Liu and Joaquín Moraga and Hendrik Süß},
journal= {arXiv preprint arXiv:2205.12326},
year = {2023}
}
Comments
43 pages; substantially extended and improved version: the paper now also settles the case of threefold singularities and many results now also cover the case of pairs, typos in abstract removed