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 p n , with n � 1. We describe all shifts of the motive of X in the complete motivic de- composition of a variety Y , which splits over the function field of X and satisfies the nilpotence principle. In particular, we prove the mo- tivic decomposability of generalized Severi-Brauer varieties X(p m ,D) of right ideals in D of reduced dimension p m , 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.