Incompressibility of Generic Orthogonal Grassmannians

Given a non-degenerate quadratic form over a field such that its maximal orthogonal grassmannian is 2-incompressible (a condition satisfied for generic quadratic forms of arbitrary dimension), we apply the theory of upper motives to show that all other 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 have characteristic 2). For any integer r with 0 ≤ r ≤ (dimφ)/2 we write Xr = Xr(φ) for the variety of r-dimensional totally isotropic subspaces of φ. For any r, the variety Xr is smooth and projective. It is geometrically connected if and only if r 6= (dimφ)/2. In particular, Xr is connected for any r if dimφ is odd. For even-dimensional φ and r = (dimφ)/2, the variety Xr is connected if and only the discriminant of φ is non-trivial. If the variety Xr is not connected, it has two connected components and they are isomorphic. In particular, the dimension ofXr is always the dimension of any its connected component. Here is a formula for the dimension, where d := dimφ: dimXr = r(r − 1)/2 + r(d− 2r). In the case where the quadratic form φ is “generic enough” (the precise condition is formulated in terms of the varietyXr with maximal r), we are going to show (see Theorems 2.1, 3.1, and 4.1) that the 2-motive of Xr is indecomposable, if we are away from the two exceptional cases described below (where the motive evidently decomposes). Each of the both exceptional cases arises only if the dimension of φ is even and the discriminant of φ is trivial. The first case is the case of r = (dimφ)/2, where the variety Xr has two connected components. Our assumption on φ ensures that the 2-motive of each component of Xr is indecomposable. Date: 23 November 2010.