Title: The $K$-theory of abelian symplectic quotients
Abstract:Let T be a compact torus and (M, ω) a Hamiltonian T -space.In a previous paper, the authors showed that the T -equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of th...Let T be a compact torus and (M, ω) a Hamiltonian T -space.In a previous paper, the authors showed that the T -equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of the symplectic quotient M//T , under certain technical conditions on the moment map.In this paper, we use equivariant Morse theory to give a method for computing the K-theory of M//T by obtaining an explicit description of the kernel of the surjection κ : K * T (M ) K * (M//T ).Our results are K-theoretic analogues of the work of Tolman and Weitsman for Borel equivariant cohomology.Further, we prove that under suitable technical conditions on the T -orbit stratification of M , there is an explicit Goresky-Kottwitz-MacPherson ("GKM") type combinatorial description of the K-theory of a Hamiltonian T -space in terms of fixed point data.Finally, we illustrate our methods by computing the ordinary K-theory of compact symplectic toric manifolds, which arise as symplectic quotients of an affine space C N by a linear torus action.Read More