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

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.

20 pages, 2 figures; v3: final version to appear in JLMS; v2: minor changes

References in corpus (5)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
• Fukaya categories and deformations
• Mirror maps equal SYZ maps for toric Calabi-Yau surfaces
• Homological mirror symmetry for the quintic 3-fold

Cited by in corpus (8)

• Lagrangian torus fibrations and homological mirror symmetry for the conifold
• Open Gromov-Witten invariants and SYZ under local conifold transitions
• Dual torus fibrations and homological mirror symmetry for A_n-singularities
• Lagrangian sections on mirrors of toric Calabi-Yau 3-folds
• SYZ transforms for immersed Lagrangian multi-sections
• Twin Lagrangian fibrations in mirror symmetry
• Homological mirror symmetry for local Calabi-Yau manifolds via SYZ
• Special Lagrangian and deformed Hermitian Yang-Mills on tropical manifold