论文标题
递归组轨道分层的几何不变
Geometric Invariants of Recursive Group Orbit Stratification
论文作者
论文摘要
在分层投影品种上所有超平面的线性截面的局部Euler障碍物和欧拉特征是奇异理论研究的关键几何不变剂。尽管它们的重要性，但总的来说，很难计算它们。在本文中，我们考虑了一种特殊类型的奇异性：递归组轨道。它们是满足某些假设的$ g_n $表示序列序列的组轨道。我们引入了一个名为$ c_ {sm} $不变的新的内在不变式，并使用它为当地的Euler障碍物和此类轨道的截面欧拉特性提供明确的公式。特别是，矩阵等级基因座是递归组轨道的示例。因此，作为应用，我们明确计算了普通，偏斜和对称等级基因座的这些几何不变。我们的方法是系统的和代数，因此适用于特征$ 0 $的代数封闭字段。此外，在复杂的环境中，我们还计算了所有三种级别基因座的交叉路口共同体结束复合物的茎欧特征。
The local Euler obstructions and the Euler characteristics of linear sections with all hyperplanes on a stratified projective variety are key geometric invariants in the study of singularity theory. Despite their importance, in general it is very hard to compute them. In this paper we consider a special type of singularity: the recursive group orbits. They are the group orbits of a sequence of $G_n$ representations $V_n$ satisfying certain assumptions. We introduce a new intrinsic invariant called the $c_{sm}$ invariant, and use it to give explicit formulas to the local Euler obstructions and the sectional Euler characteristics of such orbits. In particular, the matrix rank loci are examples of recursive group orbits. Thus as applications, we explicitly compute these geometry invariants for ordinary, skew-symmetric and symmetric rank loci. Our method is systematic and algebraic, thus works for algebraically closed field of characteristic $0$. Moreover, in the complex setting we also compute the stalk Euler characteristics of the Intersection Cohomology Sheaf complexes for all three types of rank loci.
