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

By N. Karpenko. 2009

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Abstract Let G be a semisimple affine algebraic group of inner type over a field F. We write 𝔛G 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 𝔛G. More precisely, it is known that the motive of any variety in 𝔛G 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 PGL A of automorphisms of a given central simple F-algebra A, for any variety in the class 𝔛G (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 𝔛G are p-incompressible. If A is a division algebra of degree pn for some n ≧ 0, then the list of p-incompressible varieties includes the generalized Severi–Brauer variety X(pm; A) of ideals of reduced dimension pm for m = 0, 1, …, n.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

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