Homological Mirror Symmetry for orbifold log Calabi-Yau surfaces

We construct mirror abstract Lefschetz fibrations associated to a class of surfaces with cyclic quotient singularities which we call effective. These surfaces can be obtained by contracting disjoint chains of smooth rational curves inside the anticanonical cycle $D$ of a smooth log Calabi-Yau surface $(Y,D)$ with maximal boundary and considering the result as an orbifold. The Fukaya-Seidel categories of these abstract Lefschetz fibrations admit semiorthogonal decompositions akin to the ones described via the derived special McKay correspondence of Ishii and Ueda arXiv:1104.2381v2 [math.AG]. We apply this construction to establish an equivalence at the large volume limit between the derived category of an effective orbifold log Calabi-Yau surface with points of type $\frac{1}{k}(1,1)$ and the Fukaya-Seidel category of its mirror Lefschetz fibration. We also compare the abstract construction to an explicit Landau-Ginzburg model defined by a Laurent polynomial associated to a toric degeneration in the case of the family of hypersurfaces $X_{k+1}\subset \mathbb{P}(1,1,1,k)$. The hypersurfaces $X_{k+1}$ admit a non-trivial moduli of complex structures, which we compare with an open subset of the space of symplectic structures on the total space of the mirror Landau-Ginzburg model via a mirror map built out of intrinsic quantities in a non-exact Fukaya-Seidel category.

💡 Research Summary

The paper studies homological mirror symmetry (HMS) for a class of orbifold log Calabi‑Yau (CY) surfaces that the author calls “effective.” An effective orbifold surface (X) is obtained from a smooth log CY pair ((Y,D)) with maximal boundary by contracting a collection of disjoint rational curves (E_i\subset D) whose self‑intersection numbers satisfy (E_i^2=-k_i<-2). Each contraction creates a cyclic quotient singularity of type (\frac{1}{k_i}(1,1)), and the resulting pair ((X,D_{\mathrm{orb}})) still has the same open CY variety (U=Y\setminus D=X\setminus D_{\mathrm{orb}}).

The first major result is a construction of an abstract Lefschetz fibration (w’:W’\to\mathbb C) associated to ((X,D_{\mathrm{orb}})) (Construction 2.2). The construction follows the Lekili‑Polishchuk handle‑attachment picture: for each orbifold point (p) of type (\frac{1}{n}(1,q)) one attaches (n) handles to a disjoint union of cylinders representing the components of (D_{\mathrm{orb}}). The handles corresponding to the special representations of the local group are placed exactly as in the Hacking‑Keating fibration (w:W\to\mathbb C) that mirrors the smooth pair ((Y,D)). The remaining handles, indexed by the non‑special representations, give rise to additional Lagrangian circles (\widetilde L_{p,d}) in the general fibre (F’). Consequently (F’) can be viewed as the fibre (F) of (w) stabilized by attaching (\sum_i(k_i-2)) extra Lagrangian circles. Stabilization does not change the total space up to Liouville deformation, which matches the expectation that adding a divisor on the B‑side corresponds to adding a superpotential on the same SYZ mirror.

On the B‑side, Ishii‑Ueda’s special McKay correspondence provides a semi‑orthogonal decomposition
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