Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of kappa : H-G(*)(M) --> H*(M//G) is generated by a small number of classes alpha is an element of H-G(*) (M) satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T. We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map kappa : H-G(*) (M) --> H*(M//G) is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two (CPS)-S-2, quotiented by the diagonal 2-torus action. [References: 9]
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机译：令G为以哈密顿方式作用于紧辛流形M上的紧致圆环，T为G的亚属。我们证明了kappa的核：HG（*）（M）-> H *（M // G ）是由少量类生成的，alpha是满足非常明确的限制属性的HG（*）（M）的元素。我们的主要工具是等距Kirwan映射，它是从M的G等价同调性到M被T辛还原的G / T等价同调性的自然图。我们证明了这张图是排斥性的。这是众所周知的结果的等变形式，即（非等变）Kirwan映射kappa：H-G（*）（M）-> H *（M // G）是射影。我们还计算了等变Kirwan映射的核，对在T = G的情况下归因于Tolman和Weitsman [TW]的结果进行了概括，并允许我们归纳地应用它们的方法。即使在昏暗的T = 1的情况下，这个结果也是新的。我们以一个有效的例子结束：两个（CPS）-S-2乘积的同调环，由对角2环作用表示。 [参考：9]
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