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

We study Homological Mirror Symmetry (HMS) for $A_n$-resolutions from the SYZ viewpoint. Let $X\to {\text{bC}}^2\mathrm{/}{\text{bZ}}_{n+1}$ be the crepant resolution of the $A_n$-singularity. The mirror of $X$ is given by a smoothing $\check{X}$ of ${\text{bC}}^2\mathrm{/}{\text{bZ}}_{n+1}$. 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.