• DOI:10.1007/S00229-017-0994-8
• Corpus ID: 39996134

On generic flag varieties of Spin(11) and Spin(12)

@article{Karpenko2018OnGF,
  title={On generic flag varieties of Spin(11) and Spin(12)},
  author={Nikita A. Karpenko},
  journal={manuscripta mathematica},
  year={2018},
  volume={157},
  pages={13-21},
  url={https://api.semanticscholar.org/CorpusID:39996134}
}

• N. Karpenko
• Published 1 September 2018
• Mathematics
• manuscripta mathematica

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 $$\mathop {\mathrm {CH}}\nolimits X$$CHX 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 $$G={\text {Spin}}(11)$$G=Spin(11) and $$G={\text {Spin}}(12)$$G=Spin(12) of this conjecture. On an equivalent note, we compute the… 

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 $$\mathop {\mathrm {CH}}\nolimits X$$CHX 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 $$G={\text {Spin}}(11)$$G=Spin(11) and $$G={\text {Spin}}(12)$$G=Spin(12) of this conjecture. On an equivalent note, we compute the Chow ring $$\mathop {\mathrm {CH}}\nolimits Y$$CHY 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 $$\mathop {\mathrm {CH}}\nolimits Y$$CHY and determine its order which is equal to $$16\;777\; 216$$16777216. On the other hand, we show that the Chow group $$\mathop {\mathrm {CH}}\nolimits _0Y$$CH0Y of 0-cycles on Y is torsion-free.