Gromov-Witten theory of Deligne-Mumford stacks

By D. Abramovich, T. Graber, Angelo Vistoli. 2006

519 citations recorded by the source corpus.

Given a smooth complex Deligne-Mumford stack ${\cal X}$ with a projective coarse moduli space, we introduce Gromov-Witten invariants of ${\cal X}$ and prove some of their basic properties, including the WDVV equation.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fc4dc4a9d76895073f0ee44abf391521e444d3eb

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In Geometry and Topology

Citations in the corpus

Orbifold quasimap theory (2014, methodology): “The argument requires the properness of the moduli stack Kg,n(X, β) of twisted stable maps, which is due to Abramovich and Vistoli [3].”
Orbifold techniques in degeneration formulas (2011, methodology): “or is 44 D.ABRAMOVICHANDB.FANTECHI(MAY 23, 2018) replaced by |Eq(η)|, the stabilizer of ηinside S|M|, introduced in [Li1, p. 574], [Li2, p. 203]. 5.2.3. The Chen–Ruan pairing and Theorem 0.4.1. As in [AGV], Section 6.4, or [CR], the formalism becomes a bit more elegant if one uses the Chen–Ruan pair”
Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties (2016, methodology): “To this end, we consider the following locally closed subscheme of S[r+ ∑ ki],[1],[r] by adding an open condition :”
Double ramification cycles on the moduli spaces of curves (2016, methodology): “We refer the reader to [1, 24] for the definitions of the moduli space of stable maps to (P1[r],∞).”
The Crepant Transformation Conjecture for Toric Complete Intersections (2014, background): “Let Xg,n,d denote the moduli space of degree-d stable maps to X from genus g orbifold curves with n marked points [1,2]; here d ∈ H2(|X|;Z).”
Quantum Cohomology and Periods (2011, methodology): “For the details of the subject, we refer the reader to the original articles [14, 13, 1].”
Parabolic sheaves on logarithmic schemes (2010, background): “If the logarithmic structure is generated by a single effective Cartier divisor D ⊆ X , so that M is the constant sheaf ND on D, and we take B to be 1 d ND , then X B/M is the root stack d √ (X, D) (see [1] or [6]).”
LOCALIZATION IN GROMOV-WITTEN THEORY AND ORBIFOLD GROMOV-WITTEN THEORY (2011, methodology): “without a section, then the resulting moduli space is Kg,n(X , β) in [2], and the evaluation maps take values in the rigidified inertial stack IX instead of the initial stack IX .”
Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, background): “with an isomorphism α: L → φ∗L1. Remark 3.1.2. As deﬁned, this is really a 2-category, because arrows of stacks in general have automorphisms, but because L is polarizing it is easy to see (see, e.g. [AGV08], §3.3.2) that it is equivalent to the associated category, whose morphisms are isomorphism c”
QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, background): “In the case that the curvature κG,0X of the quantum Kirwan morphism vanishes, one obtains in particular a morphism of small quantum cohomology rings κG,1X : T0QHG(X) → T0QH(X//G).”
Grothendieck Duality for Deligne-Mumford Stacks (2008, methodology): “pullback of the dualizing sheaf of its moduli space. Smooth orbifold points give a non trivial contribution that can be computed using the root construction (Cadman [Cad07], Abramovich-Graber-Vistoli [AGV06]). We compute also the dualizing sheaf of a local complete intersection proving that it is th”
Local Systems on Proper Algebraic V -manifolds (2010, background): “This has been remarked for example by Abramovich, Graber and Vistoli in [1], and was undoubtedly one of the reasons for Deligne’s comment about rational homology manifolds in [33].”