Let $f:X\rightarrow \mathbb{P}^1$ be an elliptically fibered K3 surface. Choose a coordinate on $\mathbb{P}^1$ and consider $X\backslash f^{-1}(\infty)\rightarrow \mathbb{C}$ as a Landau--Ginzburg model. What is the mirror dual of this LG model?
1 Answer
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In general, if $X$ is a compact smooth $n$-dimensional Calabi-Yau manifold, and $D\subset X$ is an ample (or numerically effective) divisor, then the mirror of $X$ is usually a degeneration of the mirror $X^\vee$ of $X$, namely it's a singular Calabi-Yau variety $Y$. It is expected that we should have the following equivalences
$D^b\mathit{Coh}(Y)\cong D^\mathit{perf}\mathcal{W}(X\setminus D);$
$\mathit{Perf}(Y)\cong D^\mathit{perf}\mathcal{F}(X\setminus D),$
where $\mathit{Perf}(Y)\subset D^b\mathit{Coh}(Y)$ is the triangulated subcategory of perfect complexes (it differs from $D^b\mathit{Coh}(Y)$ since $Y$ is singular), $\mathcal{F}(X\setminus D)$ is the Fukaya category of compact Lagrangian submanifolds in $X\setminus D$, and $\mathcal{W}(X\setminus D)$ is the wrapped Fukaya category, where certain non-compact Lagrangians are allowed as its objects.
A special case of the above expectation is stated as Conjecture 6.20 in the paper of Harder-Katzarkov: https://arxiv.org/abs/1708.01181, where $X\rightarrow\mathbb{P}^1$ is a Calabi-Yau fibered Calabi-Yau manifold, and $D$ is a union of (smooth) fibers, which is the case that you are interested in. In this case, by removing the fiber at infinity, one gets a Landau-Ginzburg model $X\setminus D\rightarrow\mathbb{C}$. Sometimes one can remove even more stuff from $X$, which may or may not affect your LG model on the categorical level. As a special case, we have the work of Seidel on fiberwise compactifications of Lefschetz fibrations: https://arxiv.org/abs/1504.06317. If you treat this LG model as your A-side, then there is another Fukaya category, namely the partially wrapped Fukaya category $\mathcal{W}_\Lambda(X\setminus D)$ with the stop $\Lambda\subset\partial_\infty(X\setminus D)$ determined by the fibration $X\setminus D\rightarrow\mathbb{C}$. For its definition, see the work of Sylvan (https://arxiv.org/abs/1604.02540) and Ganatra-Pardon-Shende (https://arxiv.org/abs/1706.03152). This category should be mirror to the so-called categorical resolution of $\mathit{Perf}(Y)$, in the sense that $\mathcal{F}(X\setminus D)$ is proper but not smooth as an $A_\infty$-category, while $\mathcal{W}_\Lambda(X\setminus D)$ is smooth with appropriate choice of $\Lambda$, so it provides a "resolution" of $\mathcal{F}(X\setminus D)$ in the sense of noncommutative geometry. From the point of view of algebraic geometry, if $Y$ contains only rational singularities, and $\widetilde{Y}\rightarrow Y$ is a usual resolution of singularities, then $D^b\mathit{Coh}(\widetilde{Y})$ provides a categorical resolution of $\mathit{Perf}(Y)$. Thus $\widetilde{Y}$ should be considered as the expected mirror of the LG model $X\setminus D\rightarrow\mathbb{C}$. For the more general case of irrational singularities, the construction of a categorical resolution is more involved, see the work of Kuznetsov-Lunts: https://arxiv.org/abs/1212.6170. You might also find it interesting to have a look at the work of Polishcuhk-Lekili on the homological mirror symmetry of nodal stacky curves: https://arxiv.org/abs/1705.06023, where the Auslander order plays the role of a categorical resolution.
