The orbifold Chow ring of toric Deligne-Mumford stacks

By L. Borisov, Linda Chen, Gregory G. Smith. 2003

355 citations recorded by the source corpus.

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6a791a24c0cc9c490b48c2b03e5bef3f8643ace7

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Citations in the corpus

Smooth toric Deligne-Mumford stacks (2010, methodology): “In [10] Borisov, Chen and Smith define toric Deligne-Mumford stacks as explicit global quotient (smooth) stacks, associated to combinatorial data called stacky fans.”
Orbifold quasimap theory (2014, background): “The GIT stack quotient X := [W /G] is a toric DM stack (in the sense of [5]), with quasi-projective coarse moduli space.”
LOCALIZATION IN GROMOV-WITTEN THEORY AND ORBIFOLD GROMOV-WITTEN THEORY (2011, methodology): “In this section, we recall the results of [8] on orbifold Chow ring.”
The nonequivariant coherent-constructible correspondence for toric stacks (2015, background): “of cones σˆ := Cone(ei |ei ∈σ) (5.3) for any σ∈Σ. Then βinduces a combinatorial isomorphism between Σ and Σ. Theˆ resulting toric stack is a toric DM stack without generic stabilizers in the sense of [BCS05]. As noted in Introduction, this is equivalent to toric orbifold in the sense of [FLTZ14]. Th”
Ehrhart polynomials and stringy Betti numbers (2005, methodology): “Let Y be the toric variety corresponding to the fan � whose maximal cones are the cones over the facets of the triangulation T . For a face F ∈ T and a lattice point v ∈ Box(F), �F is the fan associated to the stacky fan �/σ(v) defined in [ BCS ], and hence h�F is the vector whose i-th entry is the ”
Tropical geometry and correspondence theorems via toric stacks (2010, methodology): “In 2005, Borisov, Chen, and Smith introduced toric stacks [4].”
The tropicalization of the moduli space of curves (2012, neutral): “orphism group in the category of generalized cone complexes. Similar objects were named stacky fans in [BMV11, CMV13], but that term is also standard for combinatorial data associated to toric stacks [BCS05]. If σ is a cone in a ﬁnite diagram D of cones with face morphisms, the image of the open con”
The Crepant Transformation Conjecture for Toric Complete Intersections (2014, methodology): “, bm} ⊂ N of ray vectors of an extended stacky fan [11, 55].”
Quantum Cohomology and Periods (2011, methodology): “This parametrizes connected components of IX [7].”
An integral structure in quantum cohomology and mirror symmetry for toric orbifolds (2009, methodology): “Note that v(d) belongs to Box because v(d)= m∑ i=1 ({−〈Di, d〉}+ 〈Di, d〉)bi = m∑ i=1 {−〈Di, d〉}bi in N ⊗Q by the exact sequence (36).”
Orbifold quantum Riemann–Roch, Lefschetz and Serre (2005, background): “universal family to have certain properties. The required properties are proved in [2] for those Xthat satisfy Assumption 2.5.9. Many interesting stacks, for instance the toric Deligne-Mumford stacks [15], satisfy Assumption 2.5.9. Through the collective eﬀorts of many works, including [29], [53], [”
On the conjecture of King for smooth toric Deligne-Mumford stacks (2008, methodology): “It turns out that the action of G has only finite isotropy subgroups on U , and PΣ is then defined as the stack quotient [U/G], see [2].”
A deRham model for Chen-Ruan cohomology ring of Abelian orbifolds (2004, methodology): “bifolds and their hypersurfaces. The group structure of their Chen-Ruan cohomology is computed by M. Poddar [9], [10] and the Chen-Ruan ring structure for toric orbifolds is computed by Borisov et al,[2]. In [8], Parker et al computed the ring structure for the mirror quintic 3-fold. The diﬃculty of”
The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks (2009, background): “Let Σ = (N,Σ, β) be a stacky fan in the sense of Borisov-Chen-Smith [9].”