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Orbifold cohomology of torus quotients

By Rebecca Goldin, Tara S. Holm and Allen Knutson


ABSTRACT. We introduce the preorbifold cohomology ring PH ∗,⋄ T (Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the preorbifold cohomology ring is isomorphic to the orbifold cohomology ring H ∗ orb(Y/T) (as defined in [Chen-Ruan]) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that (Y) has a natural ring surjection onto H ∗ orb(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to PH ∗,⋄ T (Y) the graphical GKM calculus (as detailed in e.g. [Harada-Henriques-Holm]), and the kernel computations of [Tolman-Weitsman, Goldin]. We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. Toric orbifolds were previously calculated over Q in [Borisov-Chen-Smith]); 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 over Z in certain cases. are standard in ordinary cohomology. We show that PH ∗,⋄ T CONTENT

Year: 2012
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