The coherent-constructible correspondence for toric deligne-mumford stacks

We extend our previous work [8] on coherent-constructible correspondence for toric varieties to toric Deligne-Mumford (DM) stacks. Following Borisov et al. [3], a toric DM stack χ_Σ is described by a "stacky fan" Σ=(N,Σ,β), where N is a finitely generated abelian group and Σ is a simplicial fan in N_R = N _Z R. From Σ, we define a conical Lagrangian Λ_Σ inside the cotangent T*M_Rof the dual vector space M_Rof N _R, such that torus-equivariant, coherent sheaves on χ_Σ are equivalent to constructible sheaves on M_R with singular support in Λ_Σ. The microlocalization theorem of Nadler and the last author [18, 19] provides an algebro-geometrical description of the Fukaya category of a cotangent bundle T*M_R in terms of constructible sheaves on the base M_R. This allows us to interpret the main theorem stated earlier as an equivariant version of homological mirror symmetry for toric DM stacks.