karpenko at math.jussieu.fr

Submission: 2009, Mar 25

Let F be an arbitrary field. Let A be a central simple F-algebra. Let G be the algebraic group Aut A of automorphisms of A. Let C be the class of finite direct products of projective G-homogeneous F-varieties (the class C includes the generalized Severi-Brauer varieties of the algebra A). Let p be a positive prime integer. For any variety in C, we determine its canonical dimension at 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 of ideals of reduced dimension p^m for m in the interval [0,n]. We also determine the structure of the Chow motives with coefficients in F_p 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. An application of the above results is a proof of the hyperbolicity conjecture on orthogonal involutions.

2000 Mathematics Subject Classification: 14L17; 14C25

Keywords and Phrases: Algebraic groups, projective homogeneous varieties, Chow groups

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