2021/05/09 by Lekili, Yanki, Polishchuk, Alexander
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2105.03936
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 localising. 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 surface and are related to small toric resolutions of the singularity x1… xn+1= v1… vk.