The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties

We introduce the notion of the $\mathit{\text{Bruce-Roberts}}\mathit{\text{number}}$ 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 $\mathit{\text{Ebeling–Gusein-Zade}}\mathit{\text{index}}$ of $\omega$ along $X$, the $\mathit{\text{Milnor}}\mathit{\text{number}}$ of $\omega$ and the $\mathit{\text{Tjurina}}\mathit{\text{number}}$ 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.