On the topology of complex map-germs and a general Lê-Greuel formula

Abstract

Consider a singular holomorphic map-germ $f: (X,\underline{0}) \to (\mathbb C,0)$ where $X$ is a singular complex analytic variety in $\mathbb C^N$, and another holomorphic map-germ $g: (X,\underline{0}) \to (\mathbb C,0)$ which is "sufficiently good" relatively to $f$. We use stratified Morse theory to determine up to homeomorphism, the topology of the Milnor fiber $F_f$ out from the slice $F_{g,f}$ and the Morse data of a Morsification of the restriction of $g$ to $F_f$. This generalizes classical results for the case where $X$ is non-singular, and it provides a general formula comparing the Euler characteristics of $F_f$ and $F_{g,f}$. Restricting to the case where the singularity of $X$ at $\underline{0}$ is isolated, the formula for the difference of the Euler characteristics becomes algebraic and easily computable, generalizing in two directions the classical Lê-Greuel formula for the Milnor number of isolated complete intersection germs (ICIS): Firstly, $X$ can have an isolated singularity, and secondly $f$ can have arbitrary critical set. This unifies several known formulae in this vein: i) Lê-Greuel for ICIS of arbitrary codimension; ii) the formula relating the Milnor number of a curve with that of a function on it, and an extension of it for surfaces; iii) the formula for determinantal singularities; iv) and the one for the image Milnor number. All of these are special cases of our general formula.

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Theorems & Definitions (69)

• Lemma 1.1
• Lemma 1.2
• Corollary 1.3
• Definition 1.4
• Definition 1.5
• Proposition 1.6
• Definition 1.7
• Remark 1.8
• Definition 1.9
• Proposition 1.10