递归群轨道分层的几何不变量
Geometric Invariants of Recursive Group Orbit Stratification
摘要：
局部欧拉障碍和所有超平面都在分层射影簇上的线性截面的欧拉示性数是奇点理论研究中的重要几何不变量。尽管它们很重要，但通常很难计算它们。在本文中，我们考虑一种特殊类型的奇异性：递归群轨道。它们是满足某些假设的一系列$G_n$表示$V_n$的群轨道。我们引入了一个新的内禀不变量，称为$c_{sm}$不变量，并利用它给出了这种轨道的局部欧拉障碍和截面欧拉示性数的显式表达式.特别地，矩阵秩轨迹是递归群轨道的例子。因此，作为应用程序，我们明确计算这些几何不变的普通，反对称和对称秩轨迹。我们的方法是系统的和代数的，因此适用于特征$0$的代数闭域。此外，在复杂的设置，我们还计算了茎欧拉示性数的相交上同调层复形的所有三种类型的秩轨迹。
英文摘要：
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.
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