An-type surface singularity and nondisplaceable Lagrangian tori

We prove the existence of a one-parameter family of nondisplaceable Lagrangian tori near a linear chain of Lagrangian 2-spheres in a symplectic 4-manifold. When the symplectic structure is rational, we prove that the deformed Floer cohomology groups of these tori are nontrivial. The proof uses the idea of toric degeneration to analyze the full potential functions with bulk deformations of these tori.

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 Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó 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 Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.

Symplectic fillings of links of quotient surface singularities

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.