Local polar varieties in the geometric study of singularities
Local polar varieties in the geometric study of singularities
复制标题
奇点几何研究中的局部极变
DOI：
10.5802/afst.1582
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
B. Teissier
中科院分区：
文献类型：
--
作者：
A. G. Flores;B. Teissier
本文从惠特尼条件的观点出发，给出了复解析空间的等价性理论的几个方面。其目的是从几何、拓扑和代数的角度刻画一个标准的局部有限划分，该划分由一个约化的复解析空间$X$划分成非奇异层，且每个层上的$X$的局部几何是恒定的。地方极地品种之所以出现在标题中，是因为它们在统一观点方面发挥了核心作用。几何观点导致研究在非奇点上与$X相切的超平面的$X子集C^n上的极限方向空间，这反过来又导致认识到用来定义分层的惠特尼条件实际上是拉格朗日性质的。利用局部极簇来分析切超平面极限方向集的结构。这种结构特别有助于理解奇点与假设为约化的切锥的不同之处。局部极簇的多重性与局部拓扑不变量、局部消失的Euler-Poincar特征有关，通过一个公式证明，在奇点是约化射影簇上锥的顶点的特殊情况下，包含射影簇对偶度的Pl-ucker型公式.射影簇的对偶度用依附于簇的最小Whitney分层的Euler-Poincar特征来表示.
This text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical locally finite partition of a reduced complex analytic space $X$ into nonsingular strata with the property that the local geometry of $X$ is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given point of $X\subset \C^n$ of hyperplanes of $\C^n$ tangent to $X$ at nonsingular points, which in turn leads to the realization that the Whitney conditions, which are used to define the stratification, are in fact of a Lagrangian nature. The local polar varieties are used to analyze the structure of the set of limit directions of tangent hyperplanes. This structure helps in particular to understand how a singularity differs from its tangent cone, assumed to be reduced. The multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler-Poincar\'e characteristics, by a formula which turns out to contain, in the special case where the singularity is the vertex of the cone over a reduced projective variety, a Pl\"ucker-type formula for the degree of the dual of a projective variety. The degree of the dual of a projective variety is expressed in terms of Euler-Poincar\'e characteristics attached to the minimal Whitney stratification of the variety.
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