Homological mirror symmetry for the symmetric squares of punctured spheres

Advances in Mathematics. 2023. Vol. 418. Article 108942.

Lekili Y., Polishchuk A.

For an appropriate choice of a -grading structure, we prove that the wrapped Fukaya category of the symmetric square of a -punctured sphere, i.e. the Weinstein manifold given as the complement of  generic lines in  is quasi-equivalent to the derived category of coherent sheaves on a singular surface  constructed as the boundary of a toric Landau-Ginzburg model . We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localizing. We also provide a general homological mirror symmetry conjecture concerning all the higher symmetric powers of punctured spheres. The corresponding toric LG-models  are constructed from the combinatorics of curves on the punctured sphere and are related to small toric resolutions of the singularity .

Keywords: matrix factorizations, Homological Mirror symmetry, symmetric products of surfaces, Wrapped Fukaya category

Publication based on the results of: Commutative, non-commutative and motivic algebraic geometry, and geometry of special manifolds (2024)