Torsion in the full orbifold K-theory of abelian symplectic quotients

Abstract

Let (M, ω, ) be a Hamiltonian T -space and let H ⊆ T be a closed Lie subtorus. Under some technical hypotheses on the moment map , we prove that there is no additive torsion in the integral full orbifold K -theory of the orbifold symplectic quotient [M//H ]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T -space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S 1 -quotients of GKM spaces, have integral full orbifold K -theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

“…The ideas developed there motivated this project. In the case when M is the level set for an S-moment map on a Hamiltonian T-space X, our Lemma 4.1 is exactly [GHH,Lemma 2.2]. The remarks in the footnote following the local normal form theorem in [GHH, Section 2] also apply here.…”

“…Remark 4.3. A further discussion of the local normal form for symplectic toric orbifolds may be found in [GHH,Section 2]. The ideas developed there motivated this project.…”