Motives of unitary and orthogonal homogeneous varieties

Abstract Grothendieck–Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular, there have been many results which relate the motive of a one homogeneous variety to motives of other simpler or smaller ones (see for example [N.A. Karpenko, Cohomology of relative cellular spaces and of isotropic flag varieties, Algebra i Analiz 12 (1) (2000) 3–69. [Kar00a] ; V. Chernousov, S. Gille, A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (1) (2005) 137–159. [CGM05] ; P. Brosnan, On motivic decompositions arising from the method of Bialynicki-Birula, Invent. Math. 161 (1) (2005) 91–111. [Bro05] ; S. Nikolenko, N. Semenov, K. Zainoulline, Motivic decomposition of anisotropic varieties of type F 4 into generalized Rost motives, preprint, Max-Planck-Institut fur Mathematik, 90, 2005. [NSZ05] ; K.V. Zaĭnullin, N.S. Semenov, On the classification of projective homogeneous varieties up to motivic isomorphism, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006) 158–172, 273. [ZS06] ; B. Calmes, V. Petrov, N. Semenov, K. Zainoulline, Chow motives of twisted flag varieties, Compos. Math. 142 (4) (2006) 1063–1080. [CPSZ06] ; K. Zainoulline, Motivic decomposition of a generalized Severi–Brauer variety, arXiv: math.AG/0601666 . [Zai] ]). In this paper, we exhibit a relationship between motives of two homogeneous varieties by producing a natural rational map between them. As an application, we compute the Chow group of zero-dimensional cycles on a homogeneous variety associated to a Hermitian form.