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

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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.

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.

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Correspondence to Cyril Demarche.

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L’auteur a bénéficié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-12-BL01-0005.

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Demarche, C. Le groupe fondamental étale d’un espace homogène d’un groupe algébrique linéaire . Math. Ann. 368, 339–365 (2017). https://doi.org/10.1007/s00208-016-1465-3

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