Canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds

摘要

Let $M$ be a symplectic toric manifold acted on by a torus $mathbb{T}$. Inthis work we exhibit an explicit basis for the equivariant K-theory ring$mathcal{K}_{mathbb{T}}(M)$ which is canonically associated to a genericcomponent of the moment map. We provide a combinatorial algorithm for computingthe restrictions of the elements of this basis to the fixed point set; these,in turn, determine the ring structure of $mathcal{K}_{mathbb{T}}(M)$. Theconstruction is based on the notion of local index at a fixed point, similar tothat introduced by Guillemin and Kogan in [GK]. We apply the same techniques to exhibit an explicit basis for the equivariantcohomology ring $H_{mathbb{T}}(M; mathbb{Z})$ which is canonically associatedto a generic component of the moment map. Moreover we prove that the elementsof this basis coincide with some well-known sets of classes: the equivariantPoincar'e duals to the closures of unstable manifolds, and also the canonicalclasses introduced by Goldin and Tolman in [GT], which exist whenever themoment map is index increasing.