Nikita A. Karpenko: On generically split generic flag varieties

karpenko at

Submission: 2017, Apr 18, revised 2017, Oct 21

Let G be a split semisimple algebraic group over an arbitrary field F, let E be a G-torsor over F, and let P be a parabolic subgroup of G. The quotient variety X:=E/P, known as a flag variety, is generically split, if the parabolic subgroup P is special. It is generic, provided that the G-torsor E over F is a standard generic G'-torsor for a subfield F' of F and a split semisimple algebraic group G' over F' with G' over F isomoprhic to G. For any generically split generic flag variety X, we show that the Chow ring CH X is generated by Chern classes (of vector bundles over X). This implies that the topological filtration on the Grothendieck ring of X coincides with the computable gamma filtration. The results were already known in some cases including the case where P is a Borel subgroup. We also provide a complete classification of generically split generic flag varieties and, equivalently, of special parabolic subgroups for split simple groups.

2010 Mathematics Subject Classification: 20G15; 14C25

Keywords and Phrases: Algebraic groups; projective homogeneous varieties; Chow groups.

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