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 $X\times \{0\}$.

中文翻译：

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ICIS 函数的布鲁斯-罗伯茨数

我们将具有孤立奇点的函数萌芽的许多不变量与孤立完全交集奇点（ICIS）联系起来，概括了我们之前的结果，[17, 18]，因此我们提出了格鲁埃尔定理的一个新的相对基本的证明，即对于加权同质 ICIS Milnor 数等于 Tjurina 数。通过这个关系，我们可以证明任意点的相对对数特征变化是Cohen-Macaulay，并且不在$X\times \{0\}$中的任意点的对数特征变化是Cohen-Macaulay。