POINCARE DUALITY FOR TAUTOLOGICAL CHERN SUBRINGS OF ORTHOGONAL GRASSMANNIANS

Let X be an orthogonal grassmannian of a nondegenerate quadratic form q over a field. Let C be the subring in the Chow ring CH(X) generated by the Chern classes of the tautological vector bundle on X. We prove Poincare duality for C. For q of odd dimension, the result was already known due to an identification between C and the Chow ring of certain symplectic grassmannian. For q of even dimension, such an identification is not available.