Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties

TL;DR

This paper analyzes the structure of Chow motives of projective G-homogeneous varieties, determines their indecomposable summands, and identifies p-incompressible varieties, especially generalized Severi-Brauer varieties, over a field.

Contribution

It provides a complete description of the Chow motives of varieties in class C and determines their canonical dimensions and p-incompressibility for automorphism groups of central simple algebras.

Findings

Decomposition of Chow motives into indecomposables is explicitly described.

Canonical dimensions of varieties in class C are determined.

Identification of p-incompressible varieties, including generalized Severi-Brauer varieties.

Abstract

Let G be a semisimple affine algebraic group of inner type over a field F. We write C for the class of all finite direct products of projective G-homogeneous F-varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in C. More precisely, it is known that the motive of any variety in C decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group of automorphisms of a given central simple F-algebra A, for any variety in the class C (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in C are p-incompressible. If A is a division algebra of degree p^n for some n, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p^m; A) of ideals of reduced dimension p^m for m=0,1,...,n.