$A_{n}$-type surface singularity and nondisplaceable Lagrangian tori

TL;DR

This paper demonstrates the existence of nondisplaceable Lagrangian tori near certain singularities in symplectic 4-manifolds, using toric degeneration and Floer cohomology techniques.

Contribution

It introduces a method to construct and analyze nondisplaceable Lagrangian tori near $A_{n}$-type surface singularities in symplectic 4-manifolds.

Findings

Existence of a family of nondisplaceable Lagrangian tori near $A_{n}$-type singularities.

Nontrivial deformed Floer cohomology for these tori in rational symplectic structures.

Application of toric degeneration to compute potential functions with bulk deformations.

Abstract

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.