Thom condition and Monodromy
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914723835936768 |
|---|---|
| author | Conejero, R. Giménez Tráng, Lê Dũng Nuño-Ballesteros, J. J. |
| author_facet | Conejero, R. Giménez Tráng, Lê Dũng Nuño-Ballesteros, J. J. |
| contents | We give the definition of the Thom condition and we show that given any germ of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a geometric local monodromy without fixed points, provided that $f\in\mathfrak m_{X,x}^2$, where $\mathfrak m_{X,x}$ is the maximal ideal of $\mathcal O_{X,x}$. This result generalizes a well-known theorem of the second named author when $X$ is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that $X$ has maximal rectified homotopical depth at $x$ and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_05385 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Thom condition and Monodromy Conejero, R. Giménez Tráng, Lê Dũng Nuño-Ballesteros, J. J. Algebraic Geometry Algebraic Topology 32S40 (Primary) 32S50, 32S55 (Secondary) We give the definition of the Thom condition and we show that given any germ of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a geometric local monodromy without fixed points, provided that $f\in\mathfrak m_{X,x}^2$, where $\mathfrak m_{X,x}$ is the maximal ideal of $\mathcal O_{X,x}$. This result generalizes a well-known theorem of the second named author when $X$ is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that $X$ has maximal rectified homotopical depth at $x$ and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities. |
| title | Thom condition and Monodromy |
| topic | Algebraic Geometry Algebraic Topology 32S40 (Primary) 32S50, 32S55 (Secondary) |
| url | https://arxiv.org/abs/2104.05385 |