Partial desingularisations of quotients of nonsingular varieties and their Betti numbers
Partial desingularisations of quotients of nonsingular varieties and their Betti numbers
复制标题
非奇异簇商及其贝蒂数的部分去奇异化
DOI：
10.2307/1971369
复制
发表时间：
1985
影响因子：
4.9
通讯作者：
F. Kirwan
中科院分区：
文献类型：
--
作者：
F. Kirwan
当还原群 G 线性作用于非奇异复射影簇 X 时，我们可以使用 Mumford 的几何不变量理论定义射影“商”射影簇 X//G。如果满足每个半稳定点（适当）稳定的条件，则该商是该群的品种的开子集 Xss 的普通拓扑商。在[K]中，在此条件下得到X//G有理上同调的公式。该公式涉及 X 和 X 的各种线性部分的有理上同调，以及 G 的分类空间和 G 的某些还原子群的有理上同调。在许多有趣的例子中，[K] 中所需的条件并未得到满足。因此，问题是我们一般可以获得哪些信息。与 [K] 中考虑的良好情况相比，商 X//G 现在可以具有严重的奇点，其中唯一的奇点是由有限各向同性群引起的奇点。本文将证明，存在一种系统方法，可以沿着非奇异子变体序列放大 X，以获得具有 G 线性作用的变体 X，使得 X 的每个半稳定点都是稳定的。我们必须做出的唯一假设是，X 至少存在一个稳定点。那么 X//G 几乎是 X1/G 奇点的解决，从某种意义上说，最严重的奇点已经解决。此外，还有一个 X//G 有理上同调的公式，再次涉及 X 和 X 的某些线性部分的有理上同调，以及 G 的分类空间和 G 的一些还原子群的有理上同调（参见定理 8.14）。为了方便起见，我们自始至终假设 G 是连通的。然而，X 的构造通常是有效的，并且可以直接修改上同调公式以应用于一般情况。 X//G 的构造也可以修改以应用于 X 没有稳定点的某些情况。论文的布局如下。第 1 节回顾了几何不变量理论的基本事实，第 2 节描述了几何不变量理论与辛几何和矩图的关系。在第 3 节中，X 沿非奇异 C 不变子族爆炸的半稳定性和稳定性与 X 的半稳定性和稳定性相关，
When a reductive group G acts linearly on a nonsingular complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory. If the condition that every semistable point be (properly) stable is satisfied, this quotient is the ordinary topological quotient of an open subset Xss of the variety by the group. In [K] a formula is obtained for the rational cohomology of X//G under this condition. The formula involves the rational cohomology of X and various linear sections of X, together with the rational cohomology of the classifying spaces of G and certain reductive subgroups of G. In many interesting examples the condition required in [K] is not satisfied. Thus the question arises as to what information we can obtain in general. The quotient X//G can now have serious singularities in contrast to the good case considered in [K] where the only singularities are those caused by finite isotropy groups. It will be shown in this paper that there is a systematic way of blowing up X along a sequence of nonsingular subvarieties to obtain a variety X with a linear action of G such that every semistable point of X is stable. The only assumption that we have to make is that there exists at least one stable point of X. Then X//G is almost a resolution of singularities of X1/G, in the sense that the most serious singularities have been resolved. Moreover there is a formula for the rational cohomology of X//G again involving the rational cohomology of X and certain linear sections of X, together with the rational cohomology of the classifying spaces of G and some reductive subgroups of G (see Theorem 8.14). For convenience we shall assume throughout that G is connected. However the construction of X works in general, and it is straightforward to modify the cohomological formulas to apply to the general case. The construction of X//G can also be modified to apply in some cases when X has no stable points. The layout of the paper is as follows. Section 1 is a review of the basic facts of geometric invariant theory which will be needed and Section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map. In Section 3 semistability and stability in a blow-up of X along a nonsingular C-invariant subvariety are related to semistability and stability in X,