### Abstract

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_{R}of the dual vector space M_{R}of 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.

Original language | English (US) |
---|---|

Pages (from-to) | 914-954 |

Number of pages | 41 |

Journal | International Mathematics Research Notices |

Volume | 2014 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1 2014 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2014*(4), 914-954. https://doi.org/10.1093/imrn/rns235

}

*International Mathematics Research Notices*, vol. 2014, no. 4, pp. 914-954. https://doi.org/10.1093/imrn/rns235

**The coherent-constructible correspondence for toric deligne-mumford stacks.** / Fang, Bohan; Liu, Chiu Chu Melissa; Treumann, David; Zaslow, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The coherent-constructible correspondence for toric deligne-mumford stacks

AU - Fang, Bohan

AU - Liu, Chiu Chu Melissa

AU - Treumann, David

AU - Zaslow, Eric

PY - 2014/11/1

Y1 - 2014/11/1

N2 - 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 χΣ 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 NR = N Z R. From Σ, we define a conical Lagrangian ΛΣ inside the cotangent T*MRof the dual vector space MRof N R, such that torus-equivariant, coherent sheaves on χΣ are equivalent to constructible sheaves on MR with singular support in ΛΣ. 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.

AB - 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 χΣ 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 NR = N Z R. From Σ, we define a conical Lagrangian ΛΣ inside the cotangent T*MRof the dual vector space MRof N R, such that torus-equivariant, coherent sheaves on χΣ are equivalent to constructible sheaves on MR with singular support in ΛΣ. 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.

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U2 - 10.1093/imrn/rns235

DO - 10.1093/imrn/rns235

M3 - Article

VL - 2014

SP - 914

EP - 954

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 4

ER -