karpenko at ualberta.ca

Submission: 2017, Sep 25

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 CH(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=Spin(11) and G=Spin(12) of this conjecture. On an equivalent note, we compute the Chow ring CH(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 CH(Y) and determine its order which is equal to 16 777 216. On the other hand, we show that the Chow group of 0-cycles on Y is torsion-free.

2010 Mathematics Subject Classification: 20G15; 14C25

Keywords and Phrases: Quadratic forms over fields; algebraic groups; generic torsors; projective homogeneous varieties; Chow groups.

Full text: dvi.gz 19 k, dvi 40 k, ps.gz 708 k, pdf.gz 114 k, pdf 130 k.