Groupe de Picard et groupe de Brauer des compactifications lisses d’espaces homogènes

Abstract

Let k k be a field of characteristic zero, G G a connected linear algebraic group over k k and H H a connected closed k k -subgroup of G G . Let X X be a smooth k k -compactification of Y = G / H Y=G/H . We prove that the Galois lattice given by the geometric Picard group of X X is flasque. The result was known in the case H = 1 H=1 . We compute this Galois lattice up to addition of a permutation module. When G G is semisimple and simply connected, the result shows that the Brauer group of X X is determined by the maximal toric quotient of H H .