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.

中文翻译：

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紧逼近性质

本文介绍并研究了紧逼近性质，这是一种在复曲线或实曲线的函数域上定义的代数簇的性质，通过在欧几里德拓扑中加入逼近条件来细化弱逼近性质（及其已知的上同调障碍） . 我们证明了紧逼近性质是一个稳定的双有理不变量，与颤动兼容，并满足线性代数群的torsors 下的下降。它对许多合理连接的变体的有效性如下。一些具体的结果是：在实线性代数群的平滑紧致化的实轨迹中的平滑环，或在维数≥2{\geq 2} 的平滑三次超曲面中，可以用有理代数曲线来近似；