Abstract
Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring \(\mathop {\mathrm {CH}}\nolimits X\) onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove the new cases \(G={\text {Spin}}(11)\) and \(G={\text {Spin}}(12)\) of this conjecture. On an equivalent note, we compute the Chow ring \(\mathop {\mathrm {CH}}\nolimits Y\) of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group \(\mathop {\mathrm {CH}}\nolimits Y\) and determine its order which is equal to \(16\;777\; 216\). On the other hand, we show that the Chow group \(\mathop {\mathrm {CH}}\nolimits _0Y\) of 0-cycles on Y is torsion-free.
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Acknowledgements
I thank Alexander Vishik for productive discussions during our Canmore visit in August 2017, careful reading of the manuscript, and interesting comments.
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This work has been supported by a Discovery Grant from the National Science and Engineering Research Council of Canada.