The tight approximation property

Author:

Benoist Olivier1,Wittenberg Olivier2

Affiliation:

1. Département de Mathématiques et Applications , École Normale Supérieure , 45 rue d’Ulm, 75230 Paris Cedex 05 , France

2. Institut Galilée , Université Sorbonne Paris Nord , 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse , France

Abstract

Abstract 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 2 {\geq 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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Approximation and homotopy in regulous geometry;Compositio Mathematica;2023-11-09

2. Rational approximation of holomorphic maps;Annales de l'Institut Fourier;2022-11-28

3. On approximation of maps into real algebraic homogeneous spaces;Journal de Mathématiques Pures et Appliquées;2022-05

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