Weighted blowups and mirror symmetry for toric surfaces

The homological mirror symmetry (HMS) conjecture was proposed by Kontsevich in 1994 as an attempt to gain a deeper mathematical understanding of mirror symmetry. Since this time, many papers have confirmed various versions of HMS. This paper explores the relation between the B-model of a toric stack and the A-model of its mirror Landau-Ginzburg model which is one version of HMS. Here the B-model of the toric stack X gives rise to the derived category DbX of coherent sheaves on the stack (or equivariant sheaves on an atlas). The mirror is given by considering the complex torus with a superpotential W and constructing the derived Fukaya category Db (Fuk(( C* )n,W)). For this version of HMS, the conjecture is that these two triangulated categories are equivalent. This paper will show that for weighted blowups, both the A-model and B-model categories have natural semiorthogonal decompositions. An explicit equivalence of the right orthogonal categories will be demonstrated for the two dimensional case.