Duke Mathematical Journal

The elementary obstruction and homogeneous spaces

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

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Abstract

Let $k$ be a field of characteristic zero, and let ${\overline k}$ be an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\overline k}(X)$ for the function field of ${\overline X}=X\times_k{\overline k}$. If $X$ has a smooth $k$-point, the natural embedding of multiplicative groups ${\overline k}^*\hookrightarrow {\overline 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 ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ 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 ${\overline k}$ une clôture algébrique de $k$. Pour une $k$-variété $X$ géométriquement intègre, on note ${\overline k}(X)$ le corps des fonctions de ${\overline X}=X\times_k{\overline k}$. Si $X$ possède un $k$-point lisse, le plongement naturel de groupes multiplicatifs ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ admet une rétraction équivariante pour l'action du groupe de Galois de ${\overline k}$ sur $k$.

Dans la première partie de l'article, sur les corps locaux puis sur les corps globaux, on donne des conditions équivalentes à l'existence d'une telle rétraction équivariante. 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 $X$, dans beaucoup de cas mais pas dans tous, l'existence d'une rétraction équivariante à ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ implique l'existence d'un point $k$-rationnel sur $X$. Pour les espaces homogènes de groupes linéaires, la technique permet aussi de traiter le cas où $k$ est un corps de fonctions de deux variables sur les complexes

Article information

Source
Duke Math. J. Volume 141, Number 2 (2008), 321-364.

Dates
First available: 17 January 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1200601794

Digital Object Identifier
doi:10.1215/S0012-7094-08-14124-9

Zentralblatt MATH identifier
1135.14013

Mathematical Reviews number (MathSciNet)
MR2376817

Subjects
Primary: 14G05: Rational points 11G99: None of the above, but in this section 12G05: Galois cohomology [See also 14F22, 16Hxx, 16K50]
Secondary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10] 14F22: Brauer groups of schemes [See also 12G05, 16K50] 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] 20G99: None of the above, but in this section

Citation

Borovoi, M.; Colliot-Thélène, J.-L.; Skorobogatov, A. N. The elementary obstruction and homogeneous spaces. Duke Mathematical Journal 141 (2008), no. 2, 321--364. doi:10.1215/S0012-7094-08-14124-9. http://projecteuclid.org/euclid.dmj/1200601794.


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