Algebraic Cuts
Algebraic Cuts
复制标题
代数割
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
W. Graham
中科院分区：
文献类型：
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作者：
D. Edidin;W. Graham
令 X 为具有代数群 G 的线性化作用的射影簇。如果 X 是光滑的，并且地面场为 C，则几何不变理论商 X//G 可以用辛几何构造的商（“约简空间”Xr）来识别。这一结果归功于 Mumford、Guillemin 和 Sternberg，将几何不变量理论和辛几何联系起来。现在假设 G = T 是一个环面，为简单起见，我们将其尺寸为 1。在 [L] 中，Lerman 引入了一种称为辛切割的构造，该构造通过 T 作用构造与 X 相关的流形 Xc ，并将 Xr 嵌入作为定点轨迹 X T c 的组成部分。 Xc 中 Xr 的补体可以用 X 的开子流形来识别，因此 X c 的其他分量是 X T 的某些分量。 X c 的分量通过等变上同调中的定域定理联系起来。因此，根据 X 的知识，我们可以（通过割空间）推断出有关 Xr 的结果。例如，Lerman 使用切割证明了 Kalkman 的留数公式，该公式与 Jeffrey-KirwanWitten ([G-K]) 的定域定理密切相关。本文的目的是提出勒曼构造的代数版本，称为代数切割，它在任意地面场和可能的奇异方案上都有效。这对于从代数几何的角度研究 Xr 非常有用，使用 [E-G] 中开发的等变交集理论代替等变上同调。例如，勒曼对卡克曼公式的证明对于平滑方案有效
Let X be a projective variety with a linearized action of an algebraic group G. If X is smooth, and the ground field is C, then the geometric invariant theory quotient X//G can be identified with a quotient constructed using symplectic geometry, the “reduced space” Xr. This result, due to Mumford, Guillemin and Sternberg, connects geometric invariant theory and symplectic geometry. Suppose now that G = T is a torus, which for simplicity we will take to have dimension 1. In [L], Lerman introduced a construction called symplectic cutting, which constructs a manifold Xc related to X , with a T -action, and embeds Xr as a component of the fixed point locus X T c . The complement of Xr in Xc can be identified with an open submanifold of X , so the other components of X c are certain components of X T . The components of X c are linked by the localization theorem in equivariant cohomology. Thus, from knowledge of X one can (via the cut space) deduce results about Xr. For example, Lerman uses cutting to prove a residue formula due to Kalkman, which is closely connected to the localization theorem of Jeffrey-KirwanWitten ([G-K]). The purpose of this paper is to present an algebraic version of Lerman’s construction, called algebraic cutting, which is valid over arbitrary ground fields and for possibly singular schemes. This is useful in studying Xr from the point of view of algebraic geometry, using the equivariant intersection theory developed in [E-G] in place of equivariant cohomology. For example, Lerman’s proof of Kalkman’s formula becomes valid for smooth schemes over