Orthogonal and Symplectic Grassmannians of Division Algebras

We consider a central division algebra (over a field) endowed with a quadratic pair or with a symplectic involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. This covers a recent conjecture raised by M. Zhykhovich. The remaining related projective homogeneous varieties are 2-compressible in general. Let F be a field, n ≥ 1, D a central division F -algebra of degree 2 endowed with a quadratic pair σ (orthogonal case) or with a symplectic involution σ (symplectic case). For definitions as well as for basic facts about involutions on central simple algebras, we refer to [11]. We recall that in the characteristic ̸= 2 case the notion of quadratic pair is equivalent to the notion of orthogonal involution. For any integer i, we write Xi for the variety of isotropic (with respect to σ) right ideals in D of reduced dimension i. For any i, the variety Xi is smooth and projective. It is nonempty if and only if 0 ≤ i ≤ 2n−1 (X0 is simply SpecF ) and is equidimensional in this case. Moreover, it is geometrically integral except the orthogonal case with i = 2n−1. The variety X2n−1 in the orthogonal case is connected if and only if the discriminant of σ is nontrivial; otherwise it has two connected components. For any i, the variety Xi is a closed subvariety of the generalized Severi-Brauer variety SBi(D) – the variety of all right ideals in D of reduced dimension i. We recall that according to [10], for any r = 0, 1, . . . , n−1, the variety SB2r(D) is 2-incompressible. This means, roughly speaking, that any correspondence SB2r(D) SB2r(D) of odd multiplicity is dominant. In particular, any rational map SB2r(D) 99K SB2r(D) is dominant. The following theorem is the main result of this note. It extends to the symplectic case as well as to the characteristic 2 case a recent conjecture due to M. Zhykhovich, [16]. Theorem 1. For any r = 0, 1, . . . , n− 1, excluding r = n− 1 in the orthogonal case, the variety X2r is 2-incompressible. The proof will be given right after some preparation work. It extensively uses the notion of upper motives introduced in [10] and [9]. Example 12 shows that Theorem 1 precisely detects the types of those projective homogeneous varieties under the connected component Aut(D, σ) of the algebraic group Aut(D, σ), which are 2-incompressible in general, i.e., for any F , D and σ. Note that Aut(D, σ) is an absolutely simple adjoint affine algebraic group of type D2n−1 in the orthogonal case and C2n−1 in the symplectic case. Date: 21 May 2012. Revised: 3 December 2012.