# The tight approximation property

## TL;DR

This paper introduces the tight approximation property for algebraic varieties over function fields, refining weak approximation by including Euclidean topology conditions, and explores its invariance and applications.

## Contribution

It defines the tight approximation property, proves its stability and compatibility with fibrations, and demonstrates its validity for certain rationally connected varieties.

## Key findings

- Tight approximation is a stable birational invariant.
- It is compatible with fibrations and satisfies descent under torsors.
- Applications include approximation of loops by rational curves and weak approximation for homogeneous spaces.

## 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 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.