Orbifold cohomology for global quotients

By Barbara Fantechi, L. Goettsche. 2001

136 citations recorded by the source corpus.

For an orbifold X which is the quotient of a manifold Y by a finite group G we construct a noncommutative ring with an action of G such that the orbifold cohomology of X as defined in math.AG/0004129 by Chen and Ruan is the G invariant part. In the case thar Y is S^n for a surface S with trivial canonical class we prove that (a small modification of) the orbifold cohomology of X is naturally isomorphic to the cohomology ring of the Hilbert scheme of n points on S computed in math.AG/0012166 by Lehn and Sorger.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e837896be7748493af0d461d323567f270c96784

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In Geometry and Topology

1. E. Markman
2. Charles Vial
3. L. Fu
4. K. Costello
5. Claire Voisin
6. Zhiyu Tian
7. L. Borisov
8. Christoph Sorger

In Mathematical Physics

1. E. Markman
2. K. Costello
3. L. Fu
4. Claire Voisin
5. Charles Vial
6. Zhiyu Tian
7. Christoph Sorger
8. M. Lehn

Citations in the corpus

• Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products (2003, methodology): “The construction of Lehn-Sorger [26] (see also Fantechi-Göttsche [15]), which calculates the orbifold cohomology of SdX in terms of the ordinary cohomology of X, applies verbatim to calculate the small quantum cohomology of SdX in terms of that of X.”
• The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism (2003, result): “uan’s [9] orbifold cohomology theory. Another application of the results in this paper, also to appear in [8], is a conceptual explanation of the results of the computations of Fantechi and Go¨ttsche [10], which show that the orbifold cohomology of a symmetric product of abelian or K3 surfaces agree”
• Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces (2004, background): “[LS]. The cohomology ring was shown to be isomorphic to the orbifold cohomology of the symmetric product [ FG ].”
• The orbifold Chow ring of toric Deligne-Mumford stacks (2003, methodology): “. Therefore, we have R1π ∗f ∗(T X(Σ)) ∼= M mk=2 Lk [ZH/G] and taking the top Chern class completes the proof. Remark 6.4. The virtual classes in Proposition 6.3 are analogous to the classes c(g,h) in [14]. However, we do not need to use the language of parabolic bundles because we give an explicit d”
• On the Chow ring of certain algebraic hyper-K\ (2006, methodology): “The cohomology ring of the Hilbert scheme of a K3 surface has been computed in [14], [16].”
• Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties (2016, methodology): “This construction is completely parallel to the construction of orbifold cohomology due to Fantechi–Göttsche [26], which is further simplified in Jarvis–Kaufmann–Kimura [36].”
• Stringy K-theory and the Chern character (2005, background): “In particular, since H(X, G) is isomorphic [ FG ] to the stringy (or Chen–Ruan orbifold) cohomology H • orb ([X/G]), the stringy”
• The cup product of Hilbert schemes for K3 surfaces (2003, background): “B. Fantechi and L. Gottsche [ 5 ] have shown that our construction”