Actions of nilpotent groups on complex algebraic varieties
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910596490854400 |
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| author | Abboud, Marc |
| author_facet | Abboud, Marc |
| contents | We study nilpotent groups acting faithfully on complex algebraic varieties. We use a method of base change. For finite p-groups, we go from $k$, a number field, to a finite field in order to use counting lemmas. We show that a finite $p$-group of polynomial automorphisms of $k^d$ is isomorphic to a subgroup of GL$_d(k)$. For infinite groups, we go from $\mathbb{C}$ to $\mathbb{Z}_p$ and use p-adic analytic tools and the theory of p-adic Lie groups. We show that a finitely generated nilpotent group $H$ acting faithfully on a complex quasiprojective variety $X$ of dimension $d$ can be embedded into a $p$-adic Lie group acting faithfully and analytically on $\mathbb{Z}_p^d$; we deduce that $d$ is larger than the virtual derived length of $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_11004 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Actions of nilpotent groups on complex algebraic varieties Abboud, Marc Algebraic Geometry Dynamical Systems We study nilpotent groups acting faithfully on complex algebraic varieties. We use a method of base change. For finite p-groups, we go from $k$, a number field, to a finite field in order to use counting lemmas. We show that a finite $p$-group of polynomial automorphisms of $k^d$ is isomorphic to a subgroup of GL$_d(k)$. For infinite groups, we go from $\mathbb{C}$ to $\mathbb{Z}_p$ and use p-adic analytic tools and the theory of p-adic Lie groups. We show that a finitely generated nilpotent group $H$ acting faithfully on a complex quasiprojective variety $X$ of dimension $d$ can be embedded into a $p$-adic Lie group acting faithfully and analytically on $\mathbb{Z}_p^d$; we deduce that $d$ is larger than the virtual derived length of $H$. |
| title | Actions of nilpotent groups on complex algebraic varieties |
| topic | Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2201.11004 |