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Orbifold cohomology of torus quotients. (English) Zbl 1159.14029

From the abstract: We introduce the inertial cohomology ring \(NH^*_T(Y)\) 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 inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring \(H_{CR}^*(Y/T)\) 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 \(NH^*_T(Y)\) has a natural ring surjection onto \(H_{CR}^*(Y//T)\), where \(Y//T\) is the symplectic reduction of \(Y\) by \(T\) at a regular value of the moment map. We extend to \(NH^*_T(Y)\) the graphical GKM calculus (as detailed in e.g. M. Harada, A. Henriques and T. S. Holm [Adv. Math. 197, No. 1, 198–221 (2005; Zbl 1110.55003)]), and the kernel computations of S. Tolman and J. Weitsman [Commun. Anal. Geom. 11, No. 4, 751–773 (2003; Zbl 1087.53076)] and R. F. Goldin [Geom. Funct. Anal. 12, No. 3, 567–583 (2002; Zbl 1033.53072)].
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 L. A. Borisov and L. Chen and G. G. Smith [J. Am. Math. Soc. 18, No. 1, 193–215 (2005; Zbl 1178.14057)]; 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.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D20 Momentum maps; symplectic reduction
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:

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