Mathematical Research Letters

Volume 24 (2017)

Number 5

Towards a link theoretic characterization of smoothness

Pages: 1239 – 1262



Tommaso de Fernex (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Yu-Chao Tu (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)


A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact structure that the link inherits from its embedding in the variety may suffice to characterize smooth points among normal isolated singularities. He proves that this is the case in dimension $3$. In this paper, we use techniques from birational geometry to extend McLean’s result to a large class of higher dimensional singularities. We also introduce a more refined invariant of the link using CR geometry, and conjecture that this invariant is strong enough to characterize smoothness in full generality.


link, contact structure, minimal log discrepancy, Nash blow-up, CR structure

2010 Mathematics Subject Classification

Primary 14B05. Secondary 32S05, 32V05, 53D10.

The research of the first author was partially supported by NSF Grant DMS-1402907 and NSF FRG Grant DMS-1265285. The research of the second author was partially supported by NSF FRG Grant DMS-1265285.

Received 30 August 2016

Accepted 22 January 2017

Published 29 December 2017