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J-invariant of linear algebraic groups

Victor Petrov·Nikita Semenov·Kirill Zainoulline
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Abstract

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 motivic behaviour 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, G2- and F4-varieties) as well as provide new examples (exceptional varieties of types E6, E7 and E8). We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G.

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Victor Petrov, Nikita Semenov, Kirill Zainoulline. 2008-03-07. J-invariant of linear algebraic groups. https://arxiv.org/abs/math/0607476

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