Unitary Grassmannians of Division Algebras

Abstract

Abstract. We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [13], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of grassmannians of isotropic ideals of division algebras.Let F be a field, K/F a separable quadratic field extension, n an integer ≥ 1, and D a central division K-algebra of degree 2 n endowed with a K/F -unitary involution σ. For definitions as well as for basic facts about involutions on central simple algebras, we refer to [16].For any integer i, we write X i for the F -variety of isotropic (with respect to) For any i, the variety X i is smooth and projective. It is nonempty if and only if 0 ≤ i ≤ 2 n−1 (X 0 is simply Spec F ) in which case it is geometrically connected and has dimensionFor any i, the variety X i is a closed subvariety of the Weil transfer R K/F SB i D, where SB i D is the ith generalized Severi-Brauer variety of D -the K-variety of all right ideals in D of reduced dimension i. We recall that according to [10] (see [15] for a more recent and simple proof), for any r = 0, 1, . . . , n − 1, the variety R K/F SB 2 r D is 2-incompressible. This means, roughly speaking, that any self-correspondenceof odd multiplicity is dominant. In particular, any rational self-mapThe following theorem is the main result of this note. It extends to the unitary setting the results on orthogonal and symplectic involutions obtained in [13]. Theorem 1. For any r = 0, 1, . . . , n − 1, the variety X 2 r is 2-incompressible.