The Bruce-Roberts number of a function on a hypersurface with isolated singularity

Abstract

Let $(X,0)$ be an isolated hypersurface singularity defined by $\varphi :({\mathbb{C}}^n,0)\to (\mathbb{C},0)$ and $f:({\mathbb{C}}^n,0)\to \mathbb{C}$ such that the Bruce-Roberts number ${\mu }_{BR}(f,X)$ is finite. We first prove that ${\mu }_{BR}(f,X)=\mu (f)+\mu (\varphi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.