Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds
We study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman and Weitsman’s proof of the GKM theorem [TW] in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S ⊂ T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen–Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen–Ruan cohomology in terms of orbifold fixed point data.
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- [Br]M. Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry, in: Parameter Spaces (Warsaw, 1994), Banach Center Publ. 36, Polish Acad. Sci., Warsaw, 1996, pp. 25–44.Google Scholar
- [HT]T. Holm, S. Tolman, Integral Kirwan surjectivity for Hamiltonian T-manifolds, in preparation.Google Scholar
- [J]P. Johnson, Equivariant Gromov–Witten theory of one-dimensional stacks, Ph.D. thesis at the University of Michigan, 2009.Google Scholar
- [LMM]S. Luo, T. Matsumura, F. Moore, Moment angle complexes and big Cohen–Macaulayness, arXiv:1205.1566.Google Scholar
- [P]T. Panov, Cohomology of face rings, and torus actions, in: Surveys in Contemporary Mathematics, London Math. Soc. Lecture Note Ser., Vol. 347, Cambridge Univ. Press, Cambridge, 2008, pp. 165–201.Google Scholar
- [TW]S. Tolman, J. Weitsman, On the cohomology rings of Hamiltonian T-spaces, in: Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, Vol. 196, American Mathematical Society, Providence, RI, 1999. pp. 251–258.Google Scholar