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 XS is described by a " stacky fan" S =(N, S, ), where N is a finitely generated abelian group and S is a simplicial fan in NR = N. Z R. From S, we define a conical Lagrangian.S inside the cotangent T* MR of the dual vector space MR of NR, such that torus-equivariant, coherent sheaves on XS are equivalent to constructible sheaves on MR with singular support in.S. 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* MR in terms of constructible sheaves on the base MR. This allows us to interpret the main theorem stated earlier as an equivariant version of homological mirror symmetry for toric DM stacks.