On the Chow Groups of Quadratic Grassmannians

In this text we get a description of the Chow-ring (modulo 2) of the Grassmanian of the middle-dimensional planes on arbitrary projective quadric. This is only a first step in the computation of the, so-called, generic discrete invariant of quadrics. This generic invariant contains the ``splitting pattern'' and ``motivic decomposition type'' invariants as specializations. Our computation gives an important invariant J(Q) of the quadric Q. We formulate a conjecture describing the canonical dimension of Q in terms of J(Q).

2000 Mathematics Subject Classification: 11E04, 14C15, 14M15

Keywords and Phrases: Quadrics, Chow groups, Grassmannians, Steenrod operations

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