Cohomology Ring of Symplectic Quotients by Circle Actions

[摘要] In this article we are concerned with how to compute the cohomology ringof a symplectic quotient by a circle action using the information we haveabout the cohomology of the original manifold and some data at the fixedpoint set of the action. Our method is based on the Tolman-Weitsman theoremwhich gives a characterization of the kernel of the Kirwan map. First wecompute a generating set for the kernel of the Kirwan map for the case ofproduct of compact connected manifolds such that the cohomology ring of eachof them is generated by a degree two class. We assume the fixed point set isisolated; however the circle action only needs to be ``formally Hamiltonian''. By identifying the kernel, we obtain the cohomology ring of the symplecticquotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the resultsof Kalkman and Hausmann-Knutson are special cases of our result.

[学科分类] 数学（综合）