Localization and Specialization for Hamiltonian Torus Actions

Abstract

We consider a Hamiltonian action of n-dimensional torus, T^n, on a compact symplectic manifold (M,\omega) with d isolated fixed points. For every fixed point p there exists (though not unique) a class a_p in H^*_{T}(M; Q) such that the collection {a_p}, over all fixed points, forms a basis for H^*_{T}(M; Q) as an H^*(BT; Q) module. The map induced by the inclusion, \iota^*:H^*_{T}(M; Q) \rightarrow H^*_{T}(M^{T}; Q)= \oplus_{j=1}^{d}Q[x_1, ..., x_n] is injective. We use such classes {a_p} to give necessary and sufficient conditions for f=(f_1, ...,f_d) in \oplus_{j=1}^{d}Q[x_1, ..., x_n] to be in the image of \iota^*, i.e. to represent an equiviariant cohomology class on M. In the case when T is a circle and present these conditions explicitly. We explain how to combine this 1-dimensional solution with Chang-Skjelbred Lemma in order to obtain the result for a torus T of any dimension. Moreover, for a GKM T-manifold M our techniques give combinatorial description of H^*_{K}(M; Q), for a generic subgroup K \hookrightarrow T, even if M is not a GKM K-manifold.