GKM theory for torus actions with nonisolated fixed Points

Abstract

Let M2d be a compact symplectic manifold and T a compact n-dimensional torus. A Hamiltonian action, τ, of T on M is a GKM action if, for every p ∈ MT, the isotropy representation of T on TpM has pairwise linearly independent weights. For such an action, the image of the set of zero- and one-dimensional orbits under the projection onto M/T is aregular d-valent graph; Goresky, Kottwitz, and MacPherson have proved that the equivariant cohomology of M can be computed from the combinatorics of this graph. In this paper we define a “GKM action with nonisolated fixed points” to be an action, τ, of T on M with the property that for every connected component, F of MT and p ∈ F, the isotropy representation of T on the normal space to F at p has pairwise linearly independent weights. For such an action, we show that all components of MT are diffeomorphic and prove an analogue of the theorem above.