The tight approximation property
| Autor: | Olivier Wittenberg, Olivier Benoist |
|---|---|
| Přispěvatelé: | Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL) |
| Rok vydání: | 2019 |
| Předmět: |
Linear algebraic group
Pure mathematics Approximation property Applied Mathematics General Mathematics 010102 general mathematics Algebraic variety 01 natural sciences Mathematics - Algebraic Geometry Hypersurface 0103 physical sciences FOS: Mathematics 010307 mathematical physics Birational invariant Algebraic curve [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] 0101 mathematics Algebraic number [MATH]Mathematics [math] Algebraic Geometry (math.AG) Function field Mathematics |
| Zdroj: | Journal für die reine und angewandte Mathematik Journal für die reine und angewandte Mathematik, 2021, 776, pp.151-200. ⟨10.1515/crelle-2021-0003⟩ |
| ISSN: | 1435-5345 |
| DOI: | 10.48550/arxiv.1907.10859 |
| Popis: | This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension at least 2, can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation. Comment: 56 pages; v2: Example 5.4 and Theorem 8.10 strengthened (previous Theorem 8.10 now in Remark 8.12); v3: revised introduction (Theorem A added), Example 5.6 expanded; v4: more detailed proof of Lemma 8.4, improved introduction |
| Databáze: | OpenAIRE |
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