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 (although not unique) a class $a_p \in H^*_{T}(M; \mathbb{Q})$ such that the collection $\{a_p\}$, over all fixed points, forms a basis for $H^*_{T}(M; \mathbb{Q})$ as an $H^*(BT; \mathbb{Q})$ module. The map induced by the inclusion, $\iota^*:H^*_{T}(M; \mathbb{Q}) \rightarrow H^*_{T}(M^{T}; \mathbb{Q})= \oplus_{j=1}^{d}\mathbb{Q}[x_1, \ldots, x_n] $ is injective. We use such classes $\{a_p\}$ to give necessary and sufficient conditions for $f=(f_1, \ldots ,f_d)$ in $\oplus_{j=1}^{d}\mathbb{Q}[x_1, \ldots, x_n]$ to be in the image of $\iota^*$, i.e., to represent an equivariant 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 Goresky-Kottwitz-MacPherson (GKM) $T$-manifold $M$ our techniques give combinatorial description of $H^*_{K}(M; \mathbb{Q})$, for a generic subgroup $K \hookrightarrow T$, even if $M$ is not a GKM $K$-manifold.