Description:

Title: Homological mirror symmetry for triangle singularities

Abstract: This is a report on work in progress by UMass graduate student Ethan Zhou. A triangle singularity is a complex surface singularity associated to a discrete group acting on the Poincare disc generated by reflections in the sides of a hyperbolic triangle. If a complex singularity admits a smoothing then the nearby fiber M is naturally a symplectic manifold with boundary equipped with an exact symplectic form, the so called Milnor fiber. We prove the following version of the homological mirror symmetry conjecture of Kontsevich in this setting: the Fukaya category of the Milnor fiber of a smoothing of triangle singularity is equivalent to the derived category of holomorphic vector bundles on a singular projective surface with trivial canonical bundle. Our approach is guided by the Strominger--Yau--Zaslow interpretation of mirror symmetry in terms of dual Lagrangian torus fibrations.