Counter-examples to a conjecture of Karpenko for spin groups
Baek, Sanghoon1; Devyatov, Rostislav2
1Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701 (Republic of Korea)
2Department 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, 36 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 $G=\mathrm{Spin}(2n+1)$ 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 $G=\mathrm{Spin}(2n+1)$ 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) 
Baek, Sanghoon; Devyatov, Rostislav. Counter-examples to a conjecture of Karpenko for spin groups. Annales de l'Institut Fourier, Online first, 36 p.
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