Orbifold cohomology of torus quotients

Abstract

We introduce the inertial cohomology ring NHT*,◊(Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case where Y has a locally free action by T, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring HCR*(Y/T) (as defined in [CR]) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NHT*,◊(Y) has a natural ring surjection onto HCR*(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to NHT*,◊(Y) the graphical Goresky-Kottwitz-MacPherson (GKM) calculus (as detailed in, e.g., [HHH]) and the kernel computations of [TW] and [G1], [G2]. We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan ring has been computed for toric orbifolds, with Q-coefficients, in [BCS]; we reproduce their results over Q for all symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods) and extend them to Z-coefficients in certain cases, including weighted projective spaces