Algebraic orbifold quantum products

By D. Abramovich, T. Graber, Angelo Vistoli. 2001

174 citations recorded by the source corpus.

The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan's Gromov-Witten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbifold Workshop in Madison, Wisconsin. Following the spirit of the workshop, we give a nontechnical introduction to stacks, a general discussion of twisted stable maps, and a run-through of the issues that arise when generalizing Gromov-Witten theory to Deligne-Mumford stacks. We make a special effort to make our constructions as canonical as we can, systematically using the language of algebraic stacks. Our efforts bear some concrete fruits; in particular, we are able to define the Chen-Ruan product in degree 0 (the so called stringy cohomology of the stack) with integer coefficients. Beware - this note may be vanishing in front of your eyes: first, a somewhat less technical version intended for the workshop proceedings will shortly (we hope) be submitted. Second, a much more detailed paper is in preparation, in which all the technicalities which are swept under the rug here are discussed, as well as some cool stuff about taking roots.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ef8adef34dc57ab2b18f9e829b3bfc62e3eaaf41

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Citations in the corpus

Stringy K-theory and the Chern character (2005, background): “Similarly, we prove that the orbifold cohomology H • orb (X) of a complex (or almost complex) orbifold X is equal to its Chen–Ruan orbifold cohomology [CR1, AGV ].”
Katz's middle convolution algorithm (2006, background): “The blown-up surface (X, J) would be provided with a stack structure and the singular fibers would be twisted curves [1] [2].”
Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products (2003, methodology): “In this paper the algebraic approach to orbifold Gromov-Witten theory developed by Abramovich, Graber and Vistoli [2] is used.”
LOCALIZATION IN GROMOV-WITTEN THEORY AND ORBIFOLD GROMOV-WITTEN THEORY (2011, methodology): “The algebraic counterpart, the orbifold Gromov-Witten theory for smooth DeligneMumford (DM) stacks, was developed by Abramovich-Graber-Vistoli [1, 2].”
The orbifold Chow ring of toric Deligne-Mumford stacks (2003, methodology): “The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [ 2 ], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7] [8].”
Ehrhart polynomials and stringy Betti numbers (2005, methodology): “There is an algebraic version of orbifold cohomology, due to Abramovich, Graber and Vistoli [AGV], the so-called orbifold Chow ring.”
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory (2011, methodology): “Another advance is the Chen–Ruan theory [1, 3] of GW-invariants of orbifold target spaces, and the computation by Jarvis–Kimura of such invariants in the case when the target is the quotient of a point (or more generally a manifold) by the trivial action of a finite group.”
The Crepant Resolution Conjecture (2006, background): “By the point axiom and monodromy considerations, the only other invariant involving 1 is 〈1, D,D〉 = 1 2 .”
Orbifold quantum Riemann–Roch, Lefschetz and Serre (2005, methodology): “In [6] and [3, 4], orbifold stable maps are called twisted stable maps.”
Gromov-Witten theory of Deligne-Mumford stacks (2006, methodology): “Some mathematical reasons were discussed in [AGV].”
On the $K$-theory of smooth toric DM stacks (2005, background): “The main technical result of [BCS] connects the SR-cohomology of a smooth toric DM stack with its orbifold cohomology, see [CR] and [AGV].”