The elementary obstruction and homogeneous spaces

M. Borovoi, J. L. Colliot-Thélène, A. N. Skorobogatov

Research output: Contribution to journalArticlepeer-review


Let k be a field of characteristic zero, and let k be an algebraic closure of k. For a geometrically integral variety X over k, we write k(X) for the function field of X = X xk k. If X has a smooth k-point, the natural embedding of multiplicative groups k̄* right arrow-hooked k(X)* admits a Galois-equivariant retraction. In the first part of this article, equivalent conditions to the existence of such a retraction are given over local and then over global fields. Those conditions are expressed in terms of the Brauer group of X. In the second part of the article, we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For k local or global, and 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.

Original languageEnglish
Pages (from-to)321-364
Number of pages44
JournalDuke Mathematical Journal
Issue number2
StatePublished - 1 Feb 2008


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