Yagita’s counter-examples and beyond

A conjecture on a relationship between the Chow and Grothendieck rings for the generically twisted variety of Borel subgroups in a split semisimple group $G$, stated by the second author, has been disproved by Nobuaki Yagita in characteristic $0$ for $G=\operatorname{Spin}(2n+1)$ with $n=8$ and $n=9$. For $n=8$, the second author provided an alternative simpler proof of Yagita’s result, working in any characteristic, but failed to do so for $n=9$. This gap is filled here by involving a new ingredient—Pieri type $K$-theoretic formulas for highest orthogonal grassmannians. Besides, a similar counter-example for $n=10$ is produced. Note that the conjecture on $\operatorname{Spin}(2n+1)$ holds for $n$ up to $5$; it remains open for $n=6$, $n=7$, and every $n \geq 11$.

Keywords

algebraic groups, generic torsors, projective homogeneous varieties, Chow groups

2010 Mathematics Subject Classification

14C25, 20G15

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The work of the first author was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1901-02. This paper has been accomplished during the second author’s stay at the Max-Planck Institute for Mathematics in Bonn.

Received 12 December 2021

Received revised 16 June 2022

Accepted 28 June 2022

Published 26 April 2023