Localization and specialization for Hamiltonian torus actions

Abstract

We consider a Hamiltonian T action on a compact symplectic manifold (M,ω) with d isolated fixed points. For every fixed point p there exists a class ap ∈ H∗ T (M ;Q) such that the collection {ap}, for all fixed points, forms a basis for H∗ T (M ;Q) as an H∗(BT ;Q) module. The map induced by inclusion, ι∗ : H∗ T (M ;Q) → H∗ T (M ;Q) = ⊕j=1Q[x1, . . . , xn] is injective. We will use such classes {ap} to give necessary and sufficient conditions for f = (f1, . . . , fd) in ⊕j=1Q[x1, . . . , xn] to be in the image of ι∗, i.e. to represent an equiviariant cohomology class on M . We may recover the GKM Theorem when the one skeleton is 2-dimensional. Moreover, our techniques give combinatorial description when we restrict to a smaller torus, even though we are then no longer in GKM case.