Homological mirror symmetry for the symmetric squares of punctured spheres

Abstract

For an appropriate choice of a 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 CP2 is quasi-equivalent to the derived category of coherent sheaves on a singular surface Z2,k constructed as the boundary of a toric Landau-Ginzburg model (X2,k,w2,k). 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 (Xn,k,wn,k) are constructed from the combinatorics of curves on the punctured sphere and are related to small toric resolutions of the singularity x1…xn+1=v1…vk.