An Effective Algorithm for the Cohomology Ring of Symplectic Reductions
نویسنده
چکیده
Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of κ : H∗ G (M) → H∗(M//G) is generated by a small number of classes α ∈ H∗ G (M) satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T -equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map κ : H∗ G (M) → H∗(M//G) is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dimT = 1. We close with a worked example: the cohomology ring of the product of two CP s, quotiented by the diagonal 2-torus action.
منابع مشابه
Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces
These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand ...
متن کاملCohomology of Symplectic Reductions of Generic Coadjoint Orbits
Let Oλ be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of Oλ by the maximal torus T in K. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of Oλ, and it makes explicit the dependence on λ and a parameter in Lie(T )∗ =: t∗.
متن کاملOrbifold Cohomology of Torus Quotients
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 ) (as defined in [Chen-Ruan]) of the quotient orbifold Y/T . For Y a compact Hamiltonian T -space, we extend to orbif...
متن کاملThe Cohomology of Weight Varieties
We describe the cohomology ring of the symplectic reductions by tori of coadjoint orbits, or weight varieties. Weight varieties arise from representation theory considerations, and are temed as such because they are the symplectic analogue of the T-isotypic components (weight spaces) of irreducible representations of G. Recently, Tolman and Weitsman expressed the (ordinary) cohomology ring for ...
متن کاملThe Cohomology Rings of Abelian Symplectic Quotients
Let M be a symplectic manifold, equipped with a Hamiltonian action of a torus T . We give an explicit formula for the rational cohomology ring of the symplectic quotient M//T in terms of the cohomology ring of M and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain cases these methods enable us to show that the cohomology of the reduced space is to...
متن کامل