Jean-Louis Colliot-Thélène

Principal homogeneous spaces under flasque tori: Applications

In this paper, we define flasque tori and flasque resolutions of tori over an arbitrary base scheme (Sect. 1) and we establish the basic cohomological properties of flasque tori over a regular scheme (Sect. 2). These properties are then used in a systematic and sometimes biased manner in the study of various problems, which will now be briefly listed. In Section 3, an alternative approach to R-equivalence upon tori [S] is given. Section 4 studies the behaviour of the first and second cohomology groups of arbitrary tori over a regular local ring, when going over to the fraction field. Applications to the representation of elements by norm forms and quadratic forms are described in Sections 5 and 6. In Sections 7 and 8, we study the behaviour of the group of sections of a torus, and of the first cohomology group of a group of multiplicative type when going over from a local ring to its residue class field, or when going over from a discretely valued field to its completion. We thus recover and generalize results of Saltman [30, 311 on the Grunwald-Wang theorem and its relation with the Noether problem [35]. In Section 9, Formanek’s description [17] of the centre of the generic division ring as the function field of a certain torus provides a different route to two results of Saltman [31, 321. Let us now give more details on the contents of this paper. If U is an open set of an integral regular scheme X, the restriction map 148 OO21-8693/87

Variétés unirationelles non rationelles: Au-delà de l"exemple d"Artin et Mumford. (Non rational unirational varieties: Beyond the Artin-Mumford example)

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Patching and local-global principles for homogeneous spaces over function fields of p-adic curves

Keywords: non-rational but unirational variety ; Grothendieck invariants ; rationality ; Brauer invariants Reference CMA-ARTICLE-1989-001doi:10.1007/BF01850658 Record created on 2008-12-16, modified on 2016-08-08

Espaces principaux homogènes localement triviaux

Let F = K(X) be the function field of a smooth projective curve over a p-adic field K. To each rank one discrete valuation of F one may associate the completion Fv. Given an F-variety Y which is a homogeneous space of a connected reductive group G over F, one may wonder whether the existence of Fv-points on Y for each v is enough to ensure that Y has an F-point. In this paper we prove such a result in two cases : (i) Y is a smooth projective quadric and p is odd. (ii) The group G is the extension of a reductive group over the ring of integers of K, and Y is a principal homogeneous space of G. An essential use is made of recent patching results of Harbater, Hartmann and Krashen. There is a connection to injectivity properties of the Rost invariant and a result of Kato.

Cohomologie non ramifiée et conjecture de Hodge entière

Keywords: local triviality of principal homogeneous space ; reductive group scheme Reference CMA-ARTICLE-1992-001doi:10.1007/BF02699492 Record created on 2008-12-16, modified on 2016-08-08

The elementary obstruction and homogeneous spaces

Building upon the Bloch–Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of the second-named author implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Ojanguren and the first named author implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fibre. Résumé : En nous appuyant sur la conjecture de Bloch–Kato en K-théorie de Milnor, nous établissons un lien général entre le défaut de la conjecture de Hodge entière pour la cohomologie de degré 4 et le troisième groupe de cohomologie non ramifiée à coefficients Q/Z. Ceci permet de montrer que sur un solide 1 uniréglé le troisième groupe de cohomologie non ramifiée à coefficients Q/Z s’annule, ce que la K-théorie algébrique ne permet d’obtenir que dans certains cas. Ceci permet à l’inverse de déduire d’exemples ayant leur source en K-théorie que la conjecture de Hodge entière pour la cohomologie de degré 4 peut être en défaut pour les variétés rationnellement connexes. Pour certaines familles à un paramètre de surfaces, on établit un lien entre la conjecture de Hodge entière et l’indice de la fibre générique.

Algebraic families of nonzero elements of Shafarevich-Tate groups

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 ×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 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. Resume Soient k un corps de caracteristique nulle et k une cloture algebrique de k. Pour une k-variete X geometriquement integre, on note k(X) le corps des fonctions de X = X ×k k. Si X possede un k-point lisse, le plongement naturel de groupes multiplicatifs k ∗ ↪→ k(X)∗ admet une retraction equivariante pour l’action du groupe de Galois de k sur k. Dans la premiere partie de l’article, sur les corps locaux puis sur les corps globaux, on donne des conditions equivalentes a l’existence d’une telle retraction equivariante. Ces conditions s’expriment en terme du groupe de Brauer de la variete X. Dans la seconde partie de l’article, on considere le cas des espaces homogenes de groupes algebriques connexes, non necessairement lineaires, avec groupes d’isotropie DUKE MATHEMATICAL JOURNAL Vol. 141, No. 2, c

Groupe de Chow de codimension deux des variétés definies sur un corps de nombres: un théorème de finitude pour la torsion.

We show that there exist non-trivial families of algebraic varieties for which all the fibers above rational points (or even above points of odd degree) are torsors of abelian varieties representing nonzero elements of their Shafarevich-Tate groups.

Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane

SummaryLetX be a smooth, projective variety defined over a number field and let CH2 (X) denote the Chow group of codimension two cycles modulo rational equivalence. We show that if the cohomology groupH2(X,Ox) vanishes then the torsion subgroup of CH2 (X) is a finite group. This result covers all previous results in this direction. The hypothesisH2(X,Ox)=0 is used to lift line bundles.

Good reduction of the Brauer-Manin obstruction

We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.