Contact Structures on AR-singularity links

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.