Milnor fillable contact structures are universally tight

TL;DR

This paper proves that the canonical contact structure on links of normal complex singularities is universally tight, and demonstrates the existence of certain 3-manifolds with universally tight contact structures not derived from taut foliations.

Contribution

It establishes the universal tightness of canonical contact structures on complex singularity links and provides examples of 3-manifolds with unique tight contact structures.

Findings

Canonical contact structures are universally tight on links of normal complex singularities.

Existence of atoroidal 3-manifolds with universally tight contact structures not deformable from taut foliations.

Answers two open questions posed by Etnyre.

Abstract

We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre.