On the desingularization of the unipotent variety
On the desingularization of the unipotent variety
复制标题
论单能品种的去奇异化
DOI：
10.1007/bf01390010
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发表时间：
1976
影响因子：
3.1
通讯作者：
R. Steinberg
中科院分区：
文献类型：
--
作者：
R. Steinberg
定理1.1.设G是一个具有可分泛覆盖的（连通）半单代数群，V是G的幂幺元簇，B是G的Borel子群，W是V × G/B中所有（u，g B）的子集，使得gI vg B.然后~：W-~ V是V的去奇异化。这里V的非奇异元素就是正则元素，都是共轭的，在[16]中有详细的研究。其次，在G是单的情况下，出现了次正则元，它们在V的奇异元之间稠密，并形成一个沿着V具有其类属奇异性的余维2的单一类。这是一个克莱因奇点，一个代数曲面上的孤立有理双点，与G有相同的标号：例如，曲面奇点xy= z r+ l，通常标记为A，在G是A r型的情况下，作为V的类属奇点出现。此外，通过对基场的特征量char k的一些限制，曲面奇异性的普遍变形可以在相应的群G内自然实现。鉴于这些显著的事实（和其他），由Grothendieck提出并由Brieskorn和Tits证明（大部分在未发表的笔记中;但是，请参见[5]和[17]的最后一部分）注意力集中在V的更深的奇异性上，因此集中在n的所有纤维上。观察u以上的纤维是（G/B），由u固定的各种”标志”，或者等价地，包含u的各种Borel子群。在这篇文章中，我们得到了一些结果，这些纤维的尺寸和不可约成分的数量，后者涉及到Weyl群W的元素。在[17]中引入的基本思想是，给定（G/B）的两个分量Y，Z，在Weyl群中存在唯一的w，使得g1 B和g2 B处于姿态w，即，对于YxZ中的（gx B，g2 B）的稠密开集，g~ lg 2 ~ Bw B。这使我们能够得到一种分类的对组成部分的元素的W和组成部分的对合，并证明（。暗（G/B）。< 1/2（dim G，-r）。在这里和其他地方G。是u和r的中心化子
Theorem 1.1. Let G be a (connected) semisimple algebraic group with separable universal covering, V the variety of unipotent elements of G, B a Borel subgroup of G, and W the subset of all (u, g B) in V x G/B such that gI vg B. Then~: W-~ V is a desingularization of V.Here the nonsingular elements of V are just the regular elements, all conjugate, studied in some detail in [16]. Next, in case G is simple, come the subregular elements, dense among the singular elements of V and forming a single class of codimension 2 along which V has its generic singularity. Thig turns out to be a Kleinian singularity, an isolated rational double point on an algebraic surface, the one with the same label as that of G: for example, the surface singularity xy= z r+ l, commonly labeled A,, occurs as the generic singularity for V in case G is of type A r. Furthermore, with some restrictions on char k, the characteristic of the base field, the universal deformation of the surface singularity can be naturally achieved within the corresponding group G. In view of these remarkable facts (and others), conjectured by Grothendieck and proved by Brieskorn and Tits (mostly in unpublished notes; see, however,[5] and the last part of [17]) attention is focused on the deeper singularities of V, hence on all of the fibres of n. Observe that the fibre above u is (G/B)., the variety of" flags" fixed by u, or, equivalently, the variety of Borel subgroups containing u. In this article we obtain some results about the dimensions of these fibres and the numbers of irreducible components, relating the latter to elements of the Weyl group W. The basic idea, introduced in [17], is that given two components Y, Z of (G/B), there exists a unique w in the Weyl group such that gl B and g2 B are in the attitude w, that is, g~ lg2~ BwB, for a dense open set of (gxB, g2 B) in YxZ. This enables us to get a sort of classification of pairs of components in terms of elements of W and of components in terms of involutions and to prove that (.) dim (G/B).< 1/2 (dim G,-r). Here and elsewhere G. is the centralizer of u and r