Incompressibility of Orthogonal Grassmannians

We prove the following conjecture due to Bryant Mathews (2008). Let Q 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 ≤ (dim q)/2). If the degree of each closed point on Q is divisible by 2 and the Witt index of q over the function field of Q is equal to i, then the variety Q is 2-incompressible. Theorem 7, proved below, has been conjectured in the Ph.D. thesis [12, page 24] (a preprint with the conjecture appeared one year earlier). We start with some development of the theory of canonical dimension of general projective homogeneous varieties (which might be of independent interest). We fix a prime p. Let G be a semisimple affine algebraic group over a field F such that GE is of inner type for some finite galois field extension E/F of degree a power of p (E = F is allowed). Let X be a projective G-homogeneous F -variety. We refer to [5] for a definition and discussion of the notion of canonical p-dimension cdimpX of X . Actually, canonical p-dimension is defined in the context of more general algebraic varieties. For any irreducible smooth projective variety X , cdimpX is the minimal dimension of a closed subvariety Y ⊂ X with a 0-cycle of p-coprime degree on YF (X). Recall that a smooth projective X is p-incompressible, if it is irreducible and cdimpX = dimX . Proposition 1. For d := cdimpX, there exist a cycle class α ∈ CH XF (X) (over F (X)) of codimension d and a cycle class β ∈ CHdX (over F ) of dimension d such that the degree of the product βF (X) · α is not divisible by p. Proof. We use the Chow motives with coefficients in Fp := Z/pZ (as defined in [3, Chapter XII]) and write Ch for the Chow group CH modulo p. Let U(X) be the upper motive of X . By definition, U(X) is a direct summand of the motive M(X) of X such that Ch U(X) 6= 0. By [5, Theorem 5.1 and Proposition 5.2], U(X) is also a direct summand of M(X)(d−m), where m := dimX . The composition M(X) → U(X) → M(X)(d−m) is given by a correspondence f ∈ Ch(X ×X); the composition M(X)(d−m) → U(X) → M(X) is given by a correspondence g ∈ Chd(X × X). The composition of correspondences g ◦ f ∈ Chm(X ×X) is a projector on X such that U(X) = (X, g ◦ f). In particular, the Date: June 2011.