Contemporary Mathematics Volume 310, 2002 Algebraic orbifold quantum products Dan Abramovich, Tom Graber, and Angelo Vistoli 1. Introduction The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen andY. 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, our presentation is intended to be understandable not only to algebraic geometers, but also practitioners of the other disciplines represented there, including differen- tial geometers and mathematical physicists. We make a special effort to make our constructions as canonical as we can, systematically using the language of algebraic stacks. Our constructions are based on the theory of twisted stable maps developed in [N-V3], but require making explicit some details which were not studied in the paper [N-V3]. Apart from the pleasure we take in understanding these details, our efforts bear some concrete fruits in particular, we are able to define the Chen-Ruan stringy product in degree 0 (the so called stringy cohomology) with integer coefficients. We work over the field C of complex numbers (although our discussion works just as well over any field of characteristic 0). 2. Stacks and their moduli spaces There are two, quite different, ways in which "orbifolds" or "stacks" arise. 2.1. Groupoids in schemes. This is the way most differential geometers, as well as many algebraic geometers, are introduced to the subject, since it is in some sense concrete and geometric: one thinks about an object which is locally modeled on a quotient of a variety by the action of an algebraic group. Then one needs to define a good notion of maps between such objects - this is the difficult part of the picture. Concretely, one is given a "relation" , namely a morphism R --+ U x U, where each of the projections R--+ U is etale (or, more generally, smooth), with the extra Research of D.A. partially supported by NSF grant DMS-0070970. Research of T.G. partially supported by an NSF post-doctoral research fellowship. Research of A.V. partially supported by the University of Bologna, funds for selected research topics. © 2002 American Mathematical Society http://dx.doi.org/10.1090/conm/310/05397

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