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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2510.15825 |
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| _version_ | 1866909852964487168 |
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| author | Tráng, Lê Dũng Nuño-Ballesteros, Juan J. Seade, José |
| author_facet | Tráng, Lê Dũng Nuño-Ballesteros, Juan J. Seade, José |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the topology of complex map-germs and a general Lê-Greuel formula Tráng, Lê Dũng Nuño-Ballesteros, Juan J. Seade, José Algebraic Geometry 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. |
| title | On the topology of complex map-germs and a general Lê-Greuel formula |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.15825 |