Counter-examples to a conjecture of Karpenko for spin groups
[Contre-exemples à une conjecture de Karpenko pour les groupes spin]
Baek, Sanghoon1 ; Devyatov, Rostislav2
1 Department of Mathematical Sciences KAIST 291 Daehak-ro, Yuseong-gu Daejeon 305-701 (Republic of Korea)
2 Department of Mathematical Sciences KAIST 291 Daehak-ro, Yuseong-gu Daejeon 305-701 (Republic of Korea) Laboratory of Algebraic Geometry and its Applications, Department of Mathematics National Research University Higher School of Economics 6 Usacheva str. Moscow 119048 (Russian Federation)
Annales de l'Institut Fourier, Online first, 37 p.

Consider the canonical morphism from the Chow ring of a smooth variety X to the associated graded ring of the topological filtration on the Grothendieck ring of X. In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generically twisted flag variety X of a semisimple group G. The conjecture was first disproved by Nobuaki Yagita for G=Spin(2n+1) with n=8,9. Later, another counter-example to the conjecture was given by Karpenko and the first author for n=10. In this note, we provide an infinite family of counter-examples to Karpenko’s conjecture for any 2-power integer n greater than 4. This generalizes Yagita’s counter-example and its modification due to Karpenko for n=8.

Considérons le morphisme canonique de l’anneau de Chow d’une variété lisse X à l’anneau gradué associé à la filtration topologique sur l’anneau de Grothendieck de X. En général, ce morphisme n’est pas injectif. Cependant, Nikita Karpenko a supposé que ces deux anneaux sont isomorphes pour une variété de drapeaux génériquement tordue X d’un groupe semi-simple G. La conjecture a été réfutée pour la première fois par Nobuaki Yagita pour G=Spin(2n+1) avec n=8,9. Plus tard, un autre contre-exemple à la conjecture a été donné par Karpenko et le premier auteur pour n=10. Dans cette note, nous fournissons une famille infinie de contre-exemples à la conjecture de Karpenko pour tout entier n égal à une puissance de 2 et supérieur à 4. Ceci généralise le contre-exemple de Yagita et sa modification due à Karpenko pour n=8.

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DOI : 10.5802/aif.3725
Classification : 20G15, 14C25, 16E20
Keywords: Algebraic groups, Spin groups, generic torsors, projective homogeneous varieties, Chow rings, Grothendieck rings
Mots-clés : Groupes algébriques, Groupes spin, Torses génériques, Variétés homogènes projectives, Anneaux de Chow, Anneaux de Grothendieck

Baek, Sanghoon 1 ; Devyatov, Rostislav 2

1 Department of Mathematical Sciences KAIST 291 Daehak-ro, Yuseong-gu Daejeon 305-701 (Republic of Korea)
2 Department of Mathematical Sciences KAIST 291 Daehak-ro, Yuseong-gu Daejeon 305-701 (Republic of Korea) Laboratory of Algebraic Geometry and its Applications, Department of Mathematics National Research University Higher School of Economics 6 Usacheva str. Moscow 119048 (Russian Federation) 
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Baek, Sanghoon; Devyatov, Rostislav. Counter-examples to a conjecture of Karpenko for spin groups. Annales de l'Institut Fourier, Online first, 37 p.

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