Pedro Barbosa, Arturo Fernández-Pérez, Víctor León
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The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties
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.