15 July 2007 Orbifold cohomology of torus quotients
Rebecca Goldin, Tara S. Holm, Allen Knutson
Author Affiliations +
Duke Math. J. 139(1): 89-139 (15 July 2007). DOI: 10.1215/S0012-7094-07-13912-7


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


Download Citation

Rebecca Goldin. Tara S. Holm. Allen Knutson. "Orbifold cohomology of torus quotients." Duke Math. J. 139 (1) 89 - 139, 15 July 2007. https://doi.org/10.1215/S0012-7094-07-13912-7


Published: 15 July 2007
First available in Project Euclid: 13 July 2007

zbMATH: 1159.14029
MathSciNet: MR2322677
Digital Object Identifier: 10.1215/S0012-7094-07-13912-7

Primary: 14N35
Secondary: 14M15 , 14M25 , 53D20 , 53D45

Rights: Copyright © 2007 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.139 • No. 1 • 15 July 2007
Back to Top