Upper motives of products of projective linear groups

2011/10/03 by Charles De Clercq, De Clercq, Charles
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1110.0391

v2

arxiv created 2011/12/21 · arxiv updated 2011/12/22

Abstract

Fix a base field F, a finite field K and consider a sequence of central simple F-algebras A₁,...,A_n. In this note we provide some results toward a classification of the indecomposable motives lying in the motivic decompositions of projective homogeneous varieties under the action of PGL(A₁)x...xPGL(A_n) with coefficients in K. We give a complete classification of those motives if n=1 and derive from it the motivic dichotomy of projective linear groups. We then provide several classification results as well as counterexamples for arbitrary n showing that the situation is less rigid. These results involve a neat study of rational maps between generalized Severi-Brauer varieties which is certainly of independent interest.

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