Sufficiently generic orthogonal grassmannians

Abstract

We prove the following conjecture due to Bryant Mathews (2008). Let Qi be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1⩽i⩽m:=[(dimq)/2]). Assume that for a given i, the form q has the following property (possessed by the generic quadratic form): the degree of each closed point on Qi is divisible by 2i and the Witt index of q over the function field of Qi is equal to i. Then the variety Qi is 2-incompressible.Assuming that the form q is sufficiently close to the generic one in a different sense, we prove a stronger property of Qi saying that its Chow motive with coefficients in F2 (the finite field of 2 elements) is indecomposable. This result contrasts with recent results of Zhykhovich (2010) [21] on decomposability of the motives of incompressible twisted grassmannians.The above two main results of the paper were known for the quadric Q1 and the maximal grassmannian Qm due to the works of A. Vishik.The proofs are based on the theory of upper motives. The results allow one to compute the canonical 2-dimension of any projective homogeneous variety (i.e., orthogonal flag variety) associated to the generic quadratic form.This paper is an extended version of Karpenko (2011) [10] including the results of Karpenko (2010) [5].