2015/03/16 by Pabiniak, Milena, Sabatini, Silvia
#14M25 #19L47 #53D05 #53D20 #55N91 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1503.04730
Let M be a symplectic toric manifold acted on by a torus \mathbbT. In this work we exhibit an explicit basis for the equivariant K-theory ring K_\mathbbT(M) which is canonically associated to a generic component of the moment map. We provide a combinatorial algorithm for computing the restrictions of the elements of this basis to the fixed point set; these, in turn, determine the ring structure of K_\mathbbT(M). The construction is based on the notion of local index at a fixed point, similar to that introduced by Guillemin and Kogan in [GK]. We apply the same techniques to exhibit an explicit basis for the equivariant cohomology ring H_\mathbbT(M; ℤ) which is canonically associated to a generic component of the moment map. Moreover we prove that the elements of this basis coincide with some well-known sets of classes: the equivariant Poincaré duals to the closures of unstable manifolds, and also the canonical classes introduced by Goldin and Tolman in [GT], which exist whenever the moment map is index increasing.