Localization and Specialization for Hamiltonian Torus Actions

TL;DR

This paper develops localization and specialization techniques for Hamiltonian torus actions on symplectic manifolds, providing criteria for equivariant cohomology classes and extending combinatorial descriptions beyond GKM manifolds.

Contribution

It introduces new classes in equivariant cohomology that form a basis and offers explicit conditions for cohomology class representation, generalizing GKM techniques to broader settings.

Findings

Constructed basis classes for equivariant cohomology

Derived necessary and sufficient conditions for cohomology class images

Extended combinatorial descriptions to non-GKM manifolds

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.