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Groupe de Picard et groupe de Brauer des compactifications lisses d'espaces homog`enes

Jean-Louis Colliot-Th'el`ene·Boris `E. Kunyavskii
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Abstract

This is a thoroughly revised version of math.AG/0502516v1 (24 Feb. 2005). Let k be a field of characteristic zero. Let Y=G/H, where G is a connected linear algebraic group over k and H is a connected closed k-subgroup of G. Let X be a smooth compactification of Y over k. We prove the conjecture set forward in the previous version : the Galois-lattice given by the geometric Picard group of X is flasque. The previous version had only partial results in this direction. They were obtained at the expense of a long d'etour via local and global fields. We owe the drastic improvement to a suggestion by O. Gabber. The previous version also assumed G semisimple simply connected. We can dispense with this. The result now covers the previously known case Y=G, with G an arbitrary connected linear algebraic group. ----- Ceci est une version enti`erement r'evis'ee de math.AG/0502516v1 (24 f'ev. 2005). Soient k un corps de caract'eristique z'ero, G un k-groupe lin'eaire connexe et H un k-sous-groupe ferm'e connexe de G. Notons Y=G/H. Soit X une k-compactification lisse de Y. Dans la pr'ec'edente version, nous avancions la conjecture : le module galoisien donn'e par le groupe de Picard g'eom'etrique de X (c'est un r'eseau) est un module flasque. Nous 'etablissions des cas particuliers de cette conjecture, sous l'hypoth`ese suppl'ementaire que G est semi-simple simplement connexe, au moyen d'une r'eduction alambiqu'ee au cas des corps p-adiques. Le pr'esent texte, qui doit beaucoup à une suggestion d'O. Gabber, 'etablit la conjecture dans le cas g'en'eral.

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Jean-Louis Colliot-Th'el`ene, Boris `E. Kunyavskii. 2005-07-04. Groupe de Picard et groupe de Brauer des compactifications lisses d'espaces homog`enes. https://arxiv.org/abs/math/0502516

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