Orbifold cohomology of torus quotients

Abstract

We introduce the inertial cohomology ring ${\mathrm{NH}}_T^{*,◊}(Y)$ 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 $H_{\mathrm{CR}}^{*}(Y/T)$ (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 ${\mathrm{NH}}_T^{*,◊}(Y)$ has a natural ring surjection onto $H_{\mathrm{CR}}^{*}(Y//T)$, where $Y//T$ is the symplectic reduction of $Y$ by $T$ at a regular value of the moment map. We extend to ${\mathrm{NH}}_T^{*,◊}(Y)$ 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 $\mathbb{Q}$-coefficients, in [BCS]; we reproduce their results over $\mathbb{Q}$ for all symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods) and extend them to $\mathbb{Z}$-coefficients in certain cases, including weighted projective spaces