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By Nikita A. Karpenko


Abstract. Given a non-degenerate quadratic form over a field such that its maximal orthogonal grassmannian is 2-incompressible (a condition satisfied for generic quadratic formsofarbitrarydimension), weapplythetheoryofuppermotivestoshowthatallother orthogonal grassmannians of this quadratic form are 2-incompressible. This computes the canonical 2-dimension of any projective homogeneous variety (i.e., orthogonal flag variety) associated to the quadratic form. Moreover, we show that the Chow motives with coefficients in F2 (and therefore also in any field of characteristic 2, [2]) of those grassmannians are indecomposable. That is quite unexpected, especially after a recent result of [9] on decomposability of the motives of incompressible twisted grassmannians. In this note, we are working with the 2-motives of certain smooth projective varieties associated to quadratic forms over fields of arbitrary characteristic. We refer to [3] for notation and basic results concerning the quadratic forms. By 2-motives, we mean the Grothendieck Chow motives with coefficients in the finite field F2 as introduced in [3]. We are using the theory of upper motives conceived in [5] and [7]. Let ϕ be a non-zero non-degenerate quadratic form over a field F (which may hav

Year: 2011
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