The tight approximation property

Olivier Benoist (1, 2) , Olivier Wittenberg (1, 3)

1 CNRS - Centre National de la Recherche Scientifique
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 LAGA - Laboratoire Analyse, Géométrie et Applications

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.

Domaines: Mathématiques [math]; Géométrie algébrique [math.AG]

Olivier Benoist, Olivier Wittenberg. The tight approximation property. Journal für die reine und angewandte Mathematik, 2021, 776, pp.151-200. ⟨10.1515/crelle-2021-0003⟩. ⟨hal-02358007⟩