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# Diagramme illustrant la multiplication dans O. Les traits pleins relient deuxélémentsdeuxéléments dont le produit est nul. En pointillé le produit de ces deuxélémentsdeuxéléments donne celui situé au-dessus du trait et lafì eche indique la positivité. Par exemple f 2 f 3 = −e 1 .

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Dans tout cet article, on designe par k un corps de ca- racteristique differente de 2 et on appelle variete tout k-schema separe et de type fini.

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... The next result provides the first known "purely exceptional" example of two different anisotropic varieties with isomorphic motives. Recall that the similar result for groups of type G 2 obtained in [Bo03] provides a motivic isomorphism between quadric and an exceptional Fano variety. ...

... In particular, applying our arguments to a Pfister quadric one obtains the celebrated decomposition into Rost motives (see [KM02,Example 7.3]). For exceptional groups of type G 2 one immediately obtains the motivic decomposition of the Fano variety together with the motivic isomorphism constructed in [Bo03]. ...

This an extended version of the previous preprint dated by February 2005. We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splitting variety for a symbol in K_3^M(k)/3. We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F4. All our results hold for Chow motives with integral coefficients. Comment: 20 pages, XYPIC

... For exceptional varieties examples of motivic decompositions were provided by J.-P. Bonnet [Bo03] (varieties of type G 2 ) and by S. Nikolenko, N. Semenov, K. Zainoulline [NSZ] (varieties of type F 4 ). Observe that in all these examples the respective group G splits over the generic point of X. ...

... 7.6 Remark. In particular, we provided a uniform proof of the main results of papers [Bo03] and [NSZ], where the cases of G 2 -and F 4 -varieties were considered. ...

Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behaviour of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as well as provide new examples (exceptional varieties of types E6, E7 and E8). We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G. Comment: The paper containes 39 pages and uses XYPIC package

... The complete solution of the problem (i) is known for quadrics and Severi-Brauer varieties due to Izhboldin, Karpenko, Merkurjev, Rost, Vishik and others (see [Izh98], [Ka96], [Ka00], [Ro98], [Vi03]). Concerning (ii), the example (of dimension 5) was provided by Bonnet in [Bo03]. It deals with twisted flag varieties of type G 2 . ...

... By the result of Bonnet [Bo03] the motive of the twisted form ξ (G 2 /P 2 ) is isomorphic to the motive of ξ (G 2 /P 1 ) which is a 5-dimensional quadric. ...

We give a complete classification of anisotropic projective homogeneous varieties of dimension less than 6 up to motivic isomorphism. We give several criteria for anisotropic flag varieties of type A_n to have isomorphic motives.

... Proof. See 7.5 Observe that by the result of Bonnet [Bo03] the motives of X(1) and X(2) are isomorphic (here X(1) is a 5-dimensional quadric). ...

... The correspondence product of two cycles α = f α × g α ∈ CH(X × Y ) and β = f β × g β ∈ CH(Y × X) is given by (cf. [Bo03,Lem. 5]) ...

Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB_2(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.

... In the present paper we study certain twisted forms of a smooth hyperplane section of Gr (3,6). These twisted forms are smooth SL 1 (A)-equivariant compactifications of a Merkurjev-Suslin variety corresponding to a central simple algebra A of degree 3. On the other hand, these twisted forms are norm varieties corresponding to symbols in K M 3 /3 given by the Serre-Rost invariant g 3 . ...

... where the vertical arrows are the morphisms of scalar extension. By the Lefschetz hyperplane theorem the map ı * s restricted to Pic(Gr (3,6)) is an isomorphism. Since Pic(SB 3 (M 2 (A))) is rational (see [15] and [16] Lemma 4.3), i.e., the left vertical arrow is an isomorphism, the restriction map res * is surjective. ...

... Next we compute the total Chern class of the tangent bundle T Ds . Since D s is a hyperplane section of Gr (3,6) we have the following exact sequence: ...

We provide a complete motivic decomposition of a twisted form of a smooth hyperplane section of Gr(3,6). This variety is a norm variety corresponding to a (3,3)-symbol.

... Remark. In particular, we provided a uniform proof of the main results of papers [Bo03] and [NSZ], where the cases of G 2 -and F 4 -varieties were considered. ...

Let X be a (colimit of) smooth algebraic variety over a subfield k of C. Let K-alg(0) (X) (resp. K(top)0(X(C))) be the algebraic (resp. topological) K-theory of k (resp. complex) vector bundles over X (resp. X(C))). When K-alg(0)(X) congruent to K-top(0)(X(C)), we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider in the cases X = BG for algebraic group G over algebraically closed fields k, and X = G(k)/T-k the twisted form of flag varieties G/T for non-algebraically closed field k.

Let Gk be a split reductive group over a field k corresponding to a compact Lie group G. Let Gk be a nontrivial Gk-torsor over a field k. In this paper we study the Chow ring of Gk. For example when (G, p) = (G2, 2), we have the isomorphism CH*(Gk)(2) $\cong$ Z(2).

Chow motives. We define the category of Chow motives (over k) follow-ing [Ma68]. Fisrt, we define the category of correspondences (over k). Its objects are smooth projective varieties over k. For morphisms, called cor-respondences, we set Mor(X, Y ) := CH,(X 脳 Y ). The pseudo-abelian completion of the category of correspondences is called the category of Chow motives and is denoted by Mk. The objects of Mk are pairs (X, p), where X is a smooth projective variety and p is a projector, that is, p p = p. The motive (X, id) will be denoted by M(X). By the construction Mk is a tensor additive self-dual category. Moreover, the Chow functor CH : Vark Z-Ab (to the category of Z-graded abelian groups) factors through Mk, i. e, one has CH

Let Gk be a split reductive group over a eld k of ch(k) = 0 corresponding to a compact Lie group G. Let H; 0 (Gk;Z=p) (resp. H(G;Z=p)) be the mod p motivic (resp. singular (topo- logical)) cohomology. Then we show the isomorphism grH; 0 (Gk;Z=p) = grH(G;Z=p) H; 0 (pt:;Z=p) for G = SOn;G2;F4;E6. Here the bidegree of the right hand side cohomology is given by deg(y )=( 2; )( resp.deg(x )=( 2 1;)) for even (resp. odd) dimensional ring generators y (resp. x).