Localization and Specialization for Hamiltonian Torus Actions

Details Localization and Specialization for Hamiltonian Torus Actions Localization and Specialization for Hamiltonian Torus Actions

2010-08-05

arxiv.org/abs/1008.0900v1

Mathematics

Symplectic Geometry

25 pages

Scientific paper

We consider a Hamiltonian $T^n$ action on a compact symplectic manifold $(M,\omega)$ with $d$ isolated fixed points. For every fixed point $p$ there exists a class $a_p \in H^*_{T}(M; \bb{Q})$ such that the collection $\{a_p\}$, for all fixed points, forms a basis for $H^*_{T}(M; \bb{Q})$ as an $H^*(BT; \bb{Q})$ module. The map induced by inclusion, $\iota^*:H^*_{T}(M; \bb{Q}) \rightarrow H^*_{T}(M^{T}; \bb{Q})= \oplus_{j=1}^{d}\bb{Q}[x_1, \ldots, x_n] $ is injective. We will use such classes $\{a_p\}$ to give necessary and sufficient conditions for $f=(f_1, \ldots ,f_d)$ in $\oplus_{j=1}^{d}\bb{Q}[x_1, \ldots, x_n]$ to be in the image of $\iota^*$, 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.

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