Fibrés principaux sur les corps valués henséliens

Abstract

Let (K,v) be a valued field, Y a K-variety, G an algebraic group over K (not necessarily smooth), and f: X->Y a G-torsor over Y. We consider the induced map X(K)-->Y(K), which is continuous for the topologies deduced from the valuation. Let I denote the image of this map. We prove that if (K,v) is henselian and its completion is a separable extension, then: - I is locally closed in Y(K); - the induced surjection X(K)-->I is a principal bundle with group G(K) (also topologized by the valuation).

59 pages, in French. Minor improvements, some changes in presentation required by journal. Compactification theorem 4.2 improved (the compactification is now equivariant under a smooth group action). Accepted for publication in Algebraic Geometry (June 4, 2014)