Local mirror symmetry via SYZ (Q6634557)

The article under review proves homological mirror symmetry for a class of affine varieties arising as local models in the Gross-Siebert program. The affine varieties in question are of the form\N\[\NX_{n,m} = \lbrace z_0\cdots z_n = 1+u_1+\cdots+u_m \mid z_i\in\mathbb{C}, u_j\in\mathbb{C}^*\rbrace.\N\]\NThe main result of the paper shows that\N\[\N\mathrm{Fuk}(X_{n,m})\cong\mathrm{Coh}(X_{m,n})\N\]\Nwhere the left-hand side denotes the wrapped Fukaya category and the right-hand side denotes the category of coherent sheaves. The author also gives a description of \(X_{n,m}\) as an instance of the non-toric blowup construction of Gross-Hacking-Keel.\N\NThe spaces \(X_{n,m}\) admit Lagrangian torus fibrations, with some singular fibers, given by\N\[\N(z_0,\ldots,z_n,u_1,\ldots,u_m)\mapsto (|z_0|^2-|z_1|^2,\ldots,|z_0|^2-|z_n|^2,\log|u_1|,\ldots,\log|u_m|)\N\]\NIn the Gross-Siebert program, mirror pairs of spaces are described by integral affine manifolds with singularities, which should be thought of as the base of a singular Lagrangian torus fibration. Passing from such a space to its mirror is acheived by replacing the singular integral affine structure on the base with the dual integral affine structure. Over a smooth point in the base, such spaces admit smooth Lagrangian torus fibrations. The varieties considered in the article under arise as local models for the total space above the singular points in the base.\N\NTo prove the mirror symmetry statement, the author uses the characterization of the wrapped Fukaya category in [\textit{S. Ganatra} et al., ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}] in terms of the category of microlocal sheaves on a Lagrangian skeleton. This involves calculating the skeleton of \(X_{n,m}\), which is achieved by relating the skeleton to the Lagrangian torus fibration. A key technical ingredient in the computation are Liouville homotopies to `tailor' the singular locus of the Lagrangian torus fibration so that its image lies near a tropical hypersurface in \(\mathbb{R}^n\). The outcome of this is a description of the Liouville skeleton of \(X_{n,m}\) in terms of an FLTZ skeleton.