The Lê-Greuel formula for functions on analytic spaces

Abstract

In this article we give an extension of the Lê-Greuel formula to the general setting of function germs $(f,g)$ defined on a complex analytic variety $X$ with arbitrary singular set, where $f = (f_1,\ldots,f_k): (X,\underline{0}) \to (\mathbb{C}^k,\underline{0})$ is generically a submersion with respect to some Whitney stratification on $X$. We assume further that the dimension of the zero set $V(f)$ is larger than 0, that $f$ has the Thom $a_f$-property with respect to this stratification, and $g: (X,\underline{0}) \to (\mathbb{C},0)$ has an isolated critical point in the stratified sense, both on $X$ and on $V(f)$.