Action de groupe sur la compactification hybride
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918256319660032 |
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| author | Roy, Alexandre |
| author_facet | Roy, Alexandre |
| contents | Let $X$ be an algebraic variety over $\mathbb{C}$ and $G$ be an algebraic group acting on $X$ whose action is closed. J. Poineau defined a compactification $X^\urcorner$ of $X(\mathbb{C})$ by using hybrid Berkovich spaces. We will focus on the extension of the action of $G$ on this compactification by characterising the set $\mathcal{U} \subset X^\urcorner$ where the action is well defined. We will also show that the quotient of $\mathcal{U}$ by the action of $G$ is homeomorphic to $(X/G)^\urcorner$, the compactification of $(X/G)(\mathbb{C})$. We then apply these results to $X = \mathrm{Rat}_d$, the space of rational maps and $G = \mathrm{SL}_2$. It gives the results of C. Favre-C. Gong in a more general setting. Furthermore, we get a compactification of $\mathrm{M}_d = \mathrm{Rat}_d/\mathrm{SL}_2$ where the boundary is made of orbits of non-archimedean rational maps. The results still holds if $\mathbb{C}$ is replaced by $k$ a non-trivially valued field and complex analytic spaces by Berkovich spaces over $k$ or if $X$ is the set of stable points of a $k$-variety defined in the sense of GIT. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00201 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Action de groupe sur la compactification hybride Roy, Alexandre Algebraic Geometry Dynamical Systems Number Theory 14D06, 14G22, 14L24, 37P50 Let $X$ be an algebraic variety over $\mathbb{C}$ and $G$ be an algebraic group acting on $X$ whose action is closed. J. Poineau defined a compactification $X^\urcorner$ of $X(\mathbb{C})$ by using hybrid Berkovich spaces. We will focus on the extension of the action of $G$ on this compactification by characterising the set $\mathcal{U} \subset X^\urcorner$ where the action is well defined. We will also show that the quotient of $\mathcal{U}$ by the action of $G$ is homeomorphic to $(X/G)^\urcorner$, the compactification of $(X/G)(\mathbb{C})$. We then apply these results to $X = \mathrm{Rat}_d$, the space of rational maps and $G = \mathrm{SL}_2$. It gives the results of C. Favre-C. Gong in a more general setting. Furthermore, we get a compactification of $\mathrm{M}_d = \mathrm{Rat}_d/\mathrm{SL}_2$ where the boundary is made of orbits of non-archimedean rational maps. The results still holds if $\mathbb{C}$ is replaced by $k$ a non-trivially valued field and complex analytic spaces by Berkovich spaces over $k$ or if $X$ is the set of stable points of a $k$-variety defined in the sense of GIT. |
| title | Action de groupe sur la compactification hybride |
| topic | Algebraic Geometry Dynamical Systems Number Theory 14D06, 14G22, 14L24, 37P50 |
| url | https://arxiv.org/abs/2512.00201 |