publications

2026

1. 

   Homological Mirror Symmetry for orbifold log Calabi-Yau surfaces

   Feb 2026

   arXiv:2602.04866

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   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 \frac1k(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⊂\mathbbP(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.

   @misc{simeonov_homological_2026,
     title = {Homological {Mirror} {Symmetry} for orbifold log {Calabi}-{Yau} surfaces},
     url = {https://arxiv.org/abs/2602.04866},
     doi = {10.48550/arXiv.2602.04866},
     urldate = {2026-02-05},
     publisher = {arXiv},
     month = feb,
     year = {2026},
     note = {arXiv:2602.04866},
     keywords = {Mathematics - Symplectic Geometry, Mathematics - Algebraic Geometry},
   }