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 $n\ge 1$. 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 $X(p^m,D)$ of right ideals in $D$ of reduced dimension $p^m,m=0,1,ldots,n-1$, except the cases $p=2,m=1$ and $m=0(foranyprimep)$, where motivic indecomposability was proven by Nikita Karpenko.