Decompositions of motives of generalized Severi-Brauer varieties

TL;DR

This paper investigates the motivic decompositions of generalized Severi-Brauer varieties associated with central division algebras, providing a comprehensive description of their motives and identifying cases of decomposability and indecomposability.

Contribution

It characterizes all shifts of motives in the complete motivic decomposition of certain varieties and proves decomposability for most cases, extending previous results by Karpenko.

Findings

Describes all shifts of motives in the decomposition of varieties splitting over function fields.

Proves motivic decomposability of generalized Severi-Brauer varieties for most cases.

Identifies specific cases where motivic indecomposability holds, confirming previous results.

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>0. 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,...,n-1, except the cases p=2, m=1 and m=0 (for any prime p), where motivic indecomposability was proven by Nikita Karpenko.