The Bruce–Roberts Numbers of a Function on an ICIS

Abstract

Abstract We relate a number of invariants of a function germ with isolated singularity over an isolated complete intersection singularity (ICIS), generalizing our previous results, [17, 18], as a consequence we present a new relatively elementary proof of Greuel’s theorem that for a weighted homogeneous ICIS the Milnor number is equal to Tjurina number. With this relation, we are able to prove that the relative logarithmic characteristic variety is Cohen–Macaulay at any point and the logarithmic characteristic variety is Cohen–Macaulay at any point not in $X\times \{0\}$.