Étale cohomology of algebraic varieties over Stein compacta

• 成果类型: Article
• 署名作者: Benoist, Olivier
• 署名单位: Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01323-2
• 发表日期: 2025
• 页码: 497-536
• 关键词: holomorphic convexity; levi problem; pythagoras numbers; analytic spaces; homotopy type; 17th problem; rings; EXTENSIONS; schemes; THEOREM

摘要

We prove a comparison theorem between the & eacute;tale cohomology of algebraic varieties over Stein compacta and the singular cohomology of their analytifications. We deduce that the field of meromorphic functions in a neighborhood of a connected Stein compact subset of a normal complex space of dimension n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n$\end{document} has cohomological dimension n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n$\end{document}. As an application of Gal(C/R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm {Gal}(\mathbb{C}/\mathbb{R})$\end{document}-equivariant variants of these results, we obtain a quantitative version of Hilbert's 17th problem on compact subsets of real-analytic spaces.