Homological mirror symmetry for $A_n$-resolutions as a $T$-duality

TL;DR

This paper proves a version of Homological Mirror Symmetry for $A_n$-resolutions using SYZ duality, establishing an equivalence between Fukaya categories and derived categories of coherent sheaves.

Contribution

It constructs an explicit geometric functor via SYZ transformations that confirms Kontsevich's HMS conjecture for $A_n$-resolutions.

Findings

Constructed a functor from Fukaya category of the mirror to coherent sheaves on the resolution.

Proved the functor is an equivalence onto a full subcategory of the derived category.

Confirmed HMS conjecture for $A_n$-resolutions using SYZ approach.

Abstract

We study Homological Mirror Symmetry (HMS) for $A_n$-resolutions from the SYZ viewpoint. Let $X\to\bC^2/\bZ_{n+1}$ be the crepant resolution of the $A_n$-singularity. The mirror of $X$ is given by a smoothing $\check{X}$ of $\bC^2/\bZ_{n+1}$. Using SYZ transformations, we construct a geometric functor from a derived Fukaya category of $\check{X}$ to the derived category of coherent sheaves on $X$. We show that this is an equivalence of triangulated categories onto a full triangulated subcategory of $D^b(X)$, thus realizing Kontsevich's HMS conjecture by SYZ.