GKM theory for torus actions with non-isolated fixed points

الأصل · EN

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 pair-wise linearly independent weights. For such an action the projection of the set of zero and one-dimensional orbits onto M/T is a regular d-valent graph; and Goresky, Kottwitz and MacPherson have proved that the equivariant cohomology of M can be computed from the combinatorics of this graph. (See GKM:eqcohom.) In this paper we define a ``GKM action with non-isolated 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 pair-wise linearly independent weights. For such an action, we show that all components of Mᵗ are diffeomorphic and prove an analogue of the theorem above.