Decompositions of motives of generalized Severi-Brauer varieties

Abstract

Let p be a positive prime number and X be a Severi-Brauer variety of a central division algebra D of degree pn, with n ≥ 1. We describe all shifts of the motive of X in the complete motivic decomposition of a variety Y, which splits over the function field of X and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties X(pm, D) of right ideals in D of reduced dimension pm, m = 0, 1,.., n - 1, except the cases p = 2, m = 1 and m = 0 (for any prime p), where motivic indecomposability was proven by Nikita Karpenko.