Counter-examples to a conjecture of Karpenko via truncated Brown-Peterson cohomology

Let $G$ be a split semisimple linear algebraic group and let $X$ denote the generically twisted variety of Borel subgroups in $G$. Nikita Karpenko conjectured that the map from the Chow ring of $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. After having been verified for many $G$, the conjecture was disproved by Nobuaki Yagita for some spinor groups. Later, other counter-examples were constructed by Baek-Karpenko and Baek-Devyatov. We present a new method for constructing counter-examples that is based on the connection of the truncated Brown-Peterson cohomology with the connective K-theory. Using this method, we disprove the conjecture for new groups, including $\mathrm{Spin}_{15}$, which is now the smallest known spinor group for which the conjecture fails.

💡 Research Summary

The paper addresses a long‑standing conjecture of Nikita Karpenko which asserts that for a split semisimple linear algebraic group G over a field of characteristic 0, the canonical map from the Chow ring CH⁎(X) of the generically twisted flag variety X = E/B (where E is a generic G‑torsor and B a Borel subgroup) to the associated graded ring of the topological filtration on K₀(X) is an isomorphism. This conjecture had been verified for many classical and exceptional groups (type Aₙ, Cₙ, SOₙ, G₂, F₄, E₆) and for Spin 2n + 1 with n ≤ 5. Counter‑examples were later found for Spin 2n + 1 with n = 8, 9 by Yagita, and subsequently for all 2‑powers n ≥ 8 by Baek–Karpenko and Baek–Devyatov.

The authors introduce a new method based on truncated Brown–Peterson cohomology BP⟨2⟩⁎ and its relationship with connective K‑theory (CK⁎). BP⟨2⟩⁎ is a free oriented cohomology theory with coefficient ring ℤ_{(p)}