Valuative compactifications of analytic varieties
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911101826891776 |
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| author | Poineau, Jérôme |
| author_facet | Poineau, Jérôme |
| contents | Let $X$ be an algebraic variety over $\mathbf{C}$. We define a canonical compactification $X^{\!\urcorner}$ of the complex analytic space $X(\mathbf{C})$ by adding a Berkovich space over a trivially valued field at the boundary. The construction is functorial with respect to proper morphisms and preserves many properties, such as normality, regularity, etc. We prove a partial GAGA theorem in this setting : there is an equivalence between the categories of coherent sheaves on $X$ and $X^{\!\urcorner}$, and it induces bijections on global sections. The results still hold if $\mathbf{C}$ is replaced by a complete non-trivially valued field $k$, and complex analytic spaces by Berkovich analytic spaces over $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18643 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Valuative compactifications of analytic varieties Poineau, Jérôme Algebraic Geometry Complex Variables Number Theory 14A10, 14D06, 14G22 Let $X$ be an algebraic variety over $\mathbf{C}$. We define a canonical compactification $X^{\!\urcorner}$ of the complex analytic space $X(\mathbf{C})$ by adding a Berkovich space over a trivially valued field at the boundary. The construction is functorial with respect to proper morphisms and preserves many properties, such as normality, regularity, etc. We prove a partial GAGA theorem in this setting : there is an equivalence between the categories of coherent sheaves on $X$ and $X^{\!\urcorner}$, and it induces bijections on global sections. The results still hold if $\mathbf{C}$ is replaced by a complete non-trivially valued field $k$, and complex analytic spaces by Berkovich analytic spaces over $k$. |
| title | Valuative compactifications of analytic varieties |
| topic | Algebraic Geometry Complex Variables Number Theory 14A10, 14D06, 14G22 |
| url | https://arxiv.org/abs/2503.18643 |