Contact Structures on AR-singularity links

TL;DR

This paper explores the existence of Stein cobordisms between contact structures induced by isolated complex surface singularities, providing explicit constructions and obstructions based on singularity properties and contact invariants.

Contribution

It constructs explicit Stein cobordisms from any contact 3-manifold to canonical contact structures of proper almost rational singularities and identifies obstructions to reverse cobordisms using Ozsváth-Szabó invariants.

Findings

Constructed explicit Stein cobordism to almost rational singularities

Identified obstructions using U-filtration depth of contact invariants

Generalized contact Ozsváth-Szabó invariants for AR plumbing graphs

Abstract

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Nemethi. We also show that the construction cannot always work in the reverse direction: in fact the U-filtration depth of contact Ozsvath-Szabo invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsvath-Szabo invariants of those contact structures fillable by a AR plumbing graph, generalizing an earlier work of the first author.