Some Lê-Greuel type formulae on stratified spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916468357070848 |
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| author | Zach, Matthias |
| author_facet | Zach, Matthias |
| contents | We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$. An extension of Tib{\u a}r's Bouquet Theorem to this setup allows for a topological definition of Milnor numbers $μ(α; f)$ for each stratum $V^α$ of $X$ and we prove several formulas which compute these numbers as (alternating) sums of certain ``homological indices''. The main technical result at work in the background is a local Riemann-Roch type theorem, relating a topological obstruction to holomorphic Euler characteristics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02682 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Lê-Greuel type formulae on stratified spaces Zach, Matthias Algebraic Geometry 32S50 14Q99 32B99 We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$. An extension of Tib{\u a}r's Bouquet Theorem to this setup allows for a topological definition of Milnor numbers $μ(α; f)$ for each stratum $V^α$ of $X$ and we prove several formulas which compute these numbers as (alternating) sums of certain ``homological indices''. The main technical result at work in the background is a local Riemann-Roch type theorem, relating a topological obstruction to holomorphic Euler characteristics. |
| title | Some Lê-Greuel type formulae on stratified spaces |
| topic | Algebraic Geometry 32S50 14Q99 32B99 |
| url | https://arxiv.org/abs/2411.02682 |