Homological mirror symmetry for the symmetric squares of punctured spheres

Abstract

For an appropriate choice of a $\mathbb{Z}$-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a $(k+3)$-punctured sphere, i.e. the Weinstein manifold given as the complement of $(k+3)$ generic lines in $\mathbb{C}{P}^2$ is quasi-equivalent to the derived category of coherent sheaves on a singular surface ${\mathcal{Z}}_{2,k}$ constructed as the boundary of a toric Landau-Ginzburg model $({\mathcal{X}}_{2,k},{\mathbf{w}}_{2,k})$. We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localising. We also provide a general homological mirror symmetry conjecture concerning all the higher symmetric powers of punctured spheres. The corresponding toric LG-models $({\mathcal{X}}_{n,k},{\mathbf{w}}_{n,k})$ are constructed from the combinatorics of curves on the punctured surface and are related to small toric resolutions of the singularity $x_1\dots x_{n+1}=v_1\dots v_k$.

52 pages, 8 figures. Added Section 2.4