Geodesic
On classification of projective homogeneous varieties up to motivic isomorphism
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 13, Tome 330 (2006), pp. 158-172 Cet article a éte moissonné depuis la source Math-Net.Ru

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We give a complete classification of anisotropic projective homogeneous varieties of dimension less than 6 up to motivic isomorphism. We give several criterions for anisotropic flag varieties of type $\mathrm{A}_n$ to have isomorphic motives. 
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K. V. Zainullin; N. S. Semenov. On classification of projective homogeneous varieties up to motivic isomorphism. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 13, Tome 330 (2006), pp. 158-172. http://geodesic.mathdoc.fr/item/ZNSL_2006_330_a8/

[1] M. Artin, “Brauer–Severi varieties”, Lect. Notes Math., 917, 1982, 194–210 | MR | Zbl

[2] J.-P. Bonnet, “Un isomorphisme motivique entre deux variétés homogènes projectives sous l'action d'un groupe de type $G_2$”, Docum. Math., 8 (2003), 247–277 | MR | Zbl

[3] R. W. Carter, Simple groups of Lie type, John Wiley and Sons, 1972 | MR | Zbl

[4] V. Chernousov, A. Merkurjev, Motivic decomposition of projective homogeneous varieties and the Krull–Schmidt theorem, Preprint, 2004 | MR | Zbl

[5] B. Calmès, V. Petrov, N. Semenov, K. Zainoulline, “Chow motives of twisted flag varieties”, Compos. Math., 142:4 (2006), 1063–1080 | DOI | MR | Zbl

[6] M. Demazure, “Automorphismes et déformations des variétés de Borel”, Inv. Math., 39 (1977), 179–186 | DOI | MR | Zbl

[7] W. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin et al., 1998 | MR

[8] O. Izhboldin, “Motivic equivalence of quadratic forms”, Docum. Math., 3 (1998), 341–351 | MR | Zbl

[9] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, AMS Colloquium Publications, 44, 1998 | MR | Zbl

[10] N. A. Karpenko, “Chzhou-motivy Grotendika mnogoobrazii Severi–Brauera”, Algebra i analiz, 7:4 (1995), 196–213 | MR | Zbl

[11] N. Karpenko, “Criteria of motivic equivalence for quadratic forms and central simple algebras”, Math. Ann., 317:3 (2000), 585–611 | DOI | MR | Zbl

[12] B. Köck, “Chow motif and higher Chow theory of $G/P$”, Manuscripta Math., 70 (1991), 363–372 | DOI | MR | Zbl

[13] Yu. I. Manin, “Sootvetstviya, motivy i monoidalnye preobrazovaniya”, Matem. sb., 77:4 (1968), 475–507 | MR | Zbl

[14] A. S. Merkurev, “Nulmernye tsikly na nekotorykh involyutivnykh mnogoobraziyakh”, Zap. nauchn. semin. POMI, 227, 1995, 93–105 | MR

[15] A. S. Merkurjev, I. A. Panin, A. R. Wadsworth, “Index reduction formulas for twisted flag varieties, I”, K-Theory, 10:6 (1996), 517–596 | DOI | MR | Zbl

[16] S. Nikolenko, N. Semenov, K. Zainoulline, Motivic decomposition of anisotropic varieties of type $\mathrm{F}$ into generalized Rost motives, Preprint Max-Planck-Institut für Mathematik, 2005 | MR | Zbl

[17] I. Kersten, U. Rehmann, “General splitting of reductive groups”, Tohoku Math. J., 46 (1994), 35–70 | DOI | MR | Zbl

[18] M. Rost, The motive of a Pfister form, Preprint, 1998; http://www.math.uni-bielefeld.de/~rost

[19] R. G. Swan, “Zero cycles on quadric hypersurfaces”, Proc. Amer. Math. Soc., 107:1 (1989), 43–46 | DOI | MR | Zbl

[20] J. Tits, Classification of algebraic semisimple groups, Proc. Symp. Pure Math., 9, Amer. Math. Soc., Providence, RI, 1966 | MR

[21] B. Totaro, “Splitting fields for $\mathrm{E}_8$-torsors”, Duke Math. J., 121:3 (2004), 425–455 | DOI | MR | Zbl

[22] A. Vishik, “Motives of quadrics with applications to the theory of quadratic forms”, Lect. Notes Math., 1835, Berlin et al., 2003, 25–101 | MR