Towards a link theoretic characterization of smoothness

TL;DR

This paper explores whether the contact structure of a link can characterize smooth points of singularities, extending known results in dimension 3 to higher dimensions using birational geometry and CR geometry.

Contribution

It extends McLean's link contact structure characterization of smoothness from dimension 3 to higher dimensions with new techniques and introduces a refined link invariant based on CR geometry.

Findings

Contact structure characterizes smoothness in dimension 3.

Extension of the characterization to higher dimensions for a large class of singularities.

Proposal of a new link invariant potentially capable of fully characterizing smoothness.

Abstract

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.