Submission: 2006, Nov 3, updated 2007, Jan 30
The present paper is an essentially extended and rewritten version of the previous preprint "Chow motives of generically split varieties". The proof is different and is based on the notion of p-exceptional degrees invented by Victor Kac. This also avoids all computer-based arguments used in the previous preprint. Let $G$ be a linear algebraic group over a field $F$ and $X$ be a projective homogeneous $G$-variety such that $G$ splits over the function field of $X$. In the present paper we introduce an invariant of $G$ called $J$-invariant which characterizes the splitting properties of the Chow motive of $X$. This generalizes the respective notion invented by A.~Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, $G_2$- and $F_4$-varieties) as well as provide new ones (exceptional varieties of types $E_6$, $E_7$ and $E_8$). We also discuss applications to canonical dimensions and splitting properties of the group $G$.
2000 Mathematics Subject Classification: 14C15; 14M15
Keywords and Phrases: projective homogeneous variety, Chow motive
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