Based on Totaro’s computation of the Chow ring of classifying spaces for
orthogonal groups, we compute the Chow rings of all orthogonal Grassmannians
associated with a generic quadratic form of any dimension. This closes the gap
between the known particular cases of the quadric and the highest orthogonal
Grassmannian. We also relate two different notions of generic quadratic
forms.
Keywords
algebraic groups, projective homogeneous varieties, Chow
groups