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

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 etale 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.