Mathematische Annalen

, Volume 368, Issue 1–2, pp 339–365 | Cite as

Le groupe fondamental étale d’un espace homogène d’un groupe algébrique linéaire

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Abstract

Let X be a homogeneous space of a connected linear algebraic group G defined over an algebraic closed field k of characteristic exponent p. Let \(x\in X(k)\). We denote by H the stabilizer of x in G and we assumed that H is connected or abelian. In this text, we compute explicitely the prime-to-p-part of the étale fundamental group \(\pi _1^{\acute{\mathrm{e}}\mathrm{t}}(X,x)\) in terms of the character groups of G and H. As an application, we prove a variant of the section conjecture for homogeneous spaces.

Mathematics Subject Classification

Primary 14F35 14M17 20G20 

Résumé

Soit X un espace homogène d’un groupe algébrique linéaire connexe G sur un corps k algébriquement clos d’exposant caractéristique p. Soit \(x\in X(k)\). On désigne par H le stabilisateur de x dans G et on suppose H connexe ou abélien. Dans ce texte, on calcule explicitement la partie première à p du groupe fondamental étale \(\pi _1^{\acute{\mathrm{e}}\mathrm{t}}(X,x)\) de X, en termes des groupes de caractères de G et H. On donne une application de cette formule à une variante de la conjecture des sections pour les espaces homogènes.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ Paris 06Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris CitéParisFrance
  2. 2.Département de mathématiques et applicationsÉcole normale supérieureParisFrance

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