The elementary obstruction and homogeneous spaces
نویسنده
چکیده
Let k be a field of characteristic zero and k an algebraic closure of k. For a geometrically integral variety X over k , we write k (X) for the function field of X = X × k k. If X has a smooth k-point , the natural embedding of multiplicative groups k * ֒→ k (X) * admits a Galois-equivariant retraction. In the first part of the paper , over local and then over global fields , equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of X. In the second part of the paper , we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups , with connected geometric stabilizers. For k local or global , for such a variety X , in many situations but not all , the existence of a Galois-equivariant retraction to k * ֒→ k (X) * ensures the existence of a k-rational point on X. For homogeneous spaces of linear algebraic groups , the technique also handles the case where k is the function field of a complex surface. Résumé Soient k un corps de caractéristique nulle et k une clôture algébrique de k. Pour une k-variété X géométriquement intègre , on note k (X) le corps des fonctions de X = X × k k. Si X possède un k-point lisse , le plongement naturel de groupes multiplicatifs k * ֒→ k (X) * admet une rétractionéquivariante pour l ' action du groupe de Galois de k sur k. Dans la premì ere partie de l ' article , sur les corps locaux puis sur les corps globaux , on donne des conditionséquivalentesà l ' existence d ' une telle rétractionéquivariante. Ces conditions s ' expriment en terme du groupe de Brauer de la variété X. Dans la seconde partie de l ' article , on considère le cas des espaces homogènes de groupes algébriques connexes , non nécessairement linéaires , avec groupes d ' isotropie géométriques connexes. Pour k local ou global , pour un tel espace homogène X , 1 dans beaucoup de cas mais pas dans tous , l ' existence d ' une rétractionéquivariantè a k * ֒→ k (X) * implique l ' existence d ' un point k-rationnel sur X. Pour les espaces homogènes de groupes linéaires , …
منابع مشابه
Frames and Homogeneous Spaces
Let be a locally compact non?abelian group and be a compact subgroup of also let be a ?invariant measure on the homogeneous space . In this article, we extend the linear operator as a bounded surjective linear operator for all ?spaces with . As an application of this extension, we show that each frame for determines a frame for and each frame for arises from a frame in via...
متن کاملGeometric Formality of Homogeneous Spaces and of Biquotients
We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain sphere bundles.
متن کاملInvestigating the Impact of Lighting Educational Spaces on Learning and Academic Achievement of Elementary Students
Background In modern education, physical space is considered as a dynamic factor in students' educational activities. This study was conducted to investigating the impact of lighting educational spaces on learning and academic achievement of elementary students. Materials and Methods At a cross-sectional study (2015-2016), a total of 210 students were selected randomly as sample of study. Clust...
متن کاملHereditarily Homogeneous Generalized Topological Spaces
In this paper we study hereditarily homogeneous generalized topological spaces. Various properties of hereditarily homogeneous generalized topological spaces are discussed. We prove that a generalized topological space is hereditarily homogeneous if and only if every transposition of $X$ is a $mu$-homeomorphism on $X$.
متن کاملOn the Theory of Homogeneous Lipschitz Spaces and Campanato Spaces
In this paper the equivalence between the Campanato spaces and homogeneous Lipschitz spaces is shown through the use of elementary and constructive means. These Lipschitz spaces can be defined in terms of derivatives as well as differences. Introduction. The Campanato spaces have previously been stated by Taibleson and Weiss [13] to be duals of certain Hardy spaces. Further results will be fort...
متن کامل