论文标题
米尔诺公式的概括
A generalization of Milnor's formula
论文作者
论文摘要
我们描述了米尔诺（Milnor）公式的概括，以归因于函数$ f $的隔离性超表面奇异性，其限制$ f |（x，0）$ to任意奇异的奇异降低复杂的分析空间$（x，0）\ subset（\ mathbb c^n，0）$具有分层的孤立的singularitified sance。 Milnor编号的相应类似物，$μ_f（α; x，0）$，是地层$ \ MATHSCRS_α$ $（x，0）$ $ f |（x，0）$中的Morse关键点的数量。我们的公式根据基于nash闭合的nash修饰的衍生几何形状表示$μ_f（α; x，0）$作为同源指数。尽管该设置中的大多数拓扑方面已经了解到，但我们的考虑提供了相应的分析对应物。我们还描述了如何通过我们的公式来计算$μ_f（α; x，0）$的数字，如果闭合$ \ edline {\ Mathscrs_α} \ subset x $的闭合$ \ edine cosse the Stratum的X $是一个超曲面。
We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function $f$ whose restriction $f|(X,0)$ to an arbitrarily singular reduced complex analytic space $(X,0) \subset (\mathbb C^n,0)$ has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, $μ_f(α;X,0)$, is the number of Morse critical points in a stratum $\mathscr S_α$ of $(X,0)$ in a morsification of $f|(X,0)$. Our formula expresses $μ_f(α;X,0)$ as a homological index based on the derived geometry of the Nash modification of the closure of the stratum $\mathscr S_α$. While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers $μ_f(α;X,0)$ by means of our formula in the case where the closure $\overline{ \mathscr S_α} \subset X$ of the stratum in question is a hypersurface.
