Towards Generalizing Schubert Calculus in the Symplectic Category
Towards Generalizing Schubert Calculus in the Symplectic Category
复制标题
在辛范畴中推广舒伯特微积分
DOI：
10.4310/jsg.2009.v7.n4.a3
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
S. Tolman
中科院分区：
文献类型：
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作者：
R. Goldin;S. Tolman
本文的主要目的是将Schubert微积分中的一些思想推广到具有孤立不动点的紧致辛流形上的Hamilton环面作用的更一般的情形。给定矩映射的一个通有分支，我们对M的每个不动点p在流形M的等变上同调中定义一个典范类\alpha_p。当它们存在时，典范类形成M的等变上同调的自然基础;特别地，当M是旗簇时，这些类是等变舒伯特类。我们证明了一个规范类\alpha_p对不动点q的限制可以通过一个只依赖于矩映射值的有理函数来计算，而其他规范类对指数正好为2的点的限制。因此，结构常数可以通过类似的有理函数来计算。我们的限制公式在许多情况下是明显的正的，包括当M是一个旗流形。最后，证明了当M是GKM流形且矩映射分量是指数增的时，积分标准类的存在性。在这种情况下，我们的限制公式专门用于一个容易计算的合理和，它只依赖于GKM图。
The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component of the moment map, we define a canonical class \alpha_p in the equivariant cohomology of the manifold M for each fixed point p of M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant Schubert classes. We show that the restriction of a canonical class \alpha_p to a fixed point q can be calculated by a rational function which depends only on the value of the moment map, and the restriction of other canonical classes to points of index exactly two higher. Therefore, the structure constants can be calculated by a similar rational function. Our restriction formula is manifestly positive in many cases, including when M is a flag manifold. Finally, we prove the existence of integral canonical classes in the case that M is a GKM manifold the moment map component is index increasing. In this case, our restriction formula specializes to an easily computable rational sum which depends only on the GKM graph.