O número de Bruce-Roberts sobre uma ICIS

Abstract

In this work we study the relations between the Bruce-Roberts number, $\mu_{BR}(f,X)$, the relative Bruce-Roberts, $\mu_{BR}^{-}(f,X)$, of a function germ $f\in\mathcal{O}_{n}$ over an ICIS, $(X,0)\subset (\C^{n},0)$, and the Milnor numbers, $\mu(f)$ and $\mu(f^{-1}(0)\cap X,0)$. When $(X,0)$ is an isolated hypersurface singularity we show that $$\mu_{BR}(f,X)=\mu(f)+\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0),$$ in which $\tau(X,0)$ is the Tjurina number, and that the logarithmic characteristic variety is Cohen-Macaulay, generalizing results of Oréfice-Okamoto's Thesis. When $(X,0)$ is an ICIS we show that $$\mu_{BR}^{-}(f,X)=\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0),$$ and that the relative logarithmic characteristic variety is Cohen-Macaulay, generalizing results of Bruce and Roberts (1988).