沿复解析簇的全纯1-形式的Bruce-Roberts数
The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties
AI总结：
本文定义全纯1-形式沿复解析簇的Bruce-Roberts数，证明其可用Ebeling--Gusein-Zade指标、Milnor数和Tjurina数表示，并应用于复二维全纯叶状结构。
摘要：
我们引入了相对于复解析簇的全纯1-形式的Bruce-Roberts数概念。我们的主要结果表明，一个1-形式ω关于具有孤立奇点的复解析超曲面X的Bruce-Roberts数可以用ω沿X的Ebeling--Gusein-Zade指标、ω的Milnor数以及X的Tjurina数来表示。这一结果使我们能够重新得到沿X的全纯函数的Bruce-Roberts数的已知公式，并建立该数与径向指标以及ω沿X的局部Euler阻碍之间的联系。此外，我们给出了在复二维情形下对整体和局部全纯叶状结构的应用。
英文摘要：
We introduce the notion of the \textit{Bruce-Roberts number} for holomorphic 1-forms relative to complex analytic varieties. Our main result shows that the Bruce-Roberts number of a 1-form $ω$ with respect to a complex analytic hypersurface $X$ with an isolated singularity can be expressed in terms of the \textit{Ebeling--Gusein-Zade index} of $ω$ along $X$, the \textit{Milnor number} of $ω$ and the \textit{Tjurina number} of $X$. This result allows us to recover known formulas for the Bruce-Roberts number of a holomorphic function along $X$ and to establish connections between this number, the radial index, and the local Euler obstruction of $ω$ along $X$. Moreover, we present applications to both global and local holomorphic foliations in complex dimension two.
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