Pedro Barbosa, Arturo Fernández-Pérez, Víctor León
{"title":"The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties","authors":"Pedro Barbosa, Arturo Fernández-Pérez, Víctor León","doi":"arxiv-2409.01237","DOIUrl":null,"url":null,"abstract":"We introduce the notion of the \\textit{Bruce-Roberts number} for holomorphic\n1-forms relative to complex analytic varieties. Our main result shows that the\nBruce-Roberts number of a 1-form $\\omega$ with respect to a complex analytic\nhypersurface $X$ with an isolated singularity can be expressed in terms of the\n\\textit{Ebeling--Gusein-Zade index} of $\\omega$ along $X$, the \\textit{Milnor\nnumber} of $\\omega$ and the \\textit{Tjurina number} of $X$. This result allows\nus to recover known formulas for the Bruce-Roberts number of a holomorphic\nfunction along $X$ and to establish connections between this number, the radial\nindex, and the local Euler obstruction of $\\omega$ along $X$. Moreover, we\npresent applications to both global and local holomorphic foliations in complex\ndimension two.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01237","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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 $\omega$ 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 $\omega$ along $X$, the \textit{Milnor
number} of $\omega$ 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 $\omega$ along $X$. Moreover, we
present applications to both global and local holomorphic foliations in complex
dimension two.