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 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.
| 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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- Mathematics(all)
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The coherent-constructible correspondence for toric deligne-mumford stacks. / Fang, Bohan; Liu, Chiu Chu Melissa; Treumann, David; Zaslow, Eric.
In: International Mathematics Research Notices, Vol. 2014, No. 4, 01.11.2014, p. 914-954.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 -