Nikita A. Karpenko: Canonical dimension of orthogonal groups

Submission: 2004, Jul 26

We prove Berhuy-Reichstein's conjecture on the canonical dimension of orthogonal groups showing that for any integer n>0, the canonical dimension of SO_{2n+1} and of SO_{2n+2} is equal to n(n+1)/2. More precisely, for a given (2n+1)-dimensional quadratic form q defined over an arbitrary field F (of characteristic different from 2), we establish certain property of the correspondences on the orthogonal grassmanian X of n-dimensional totally isotropic subspaces of q, provided that the degree over F of any finite splitting field of q is divisible by 2^n; this property allows to prove that the function field of X has the minimal transcendence degree among all generic splitting fields of q.

2000 Mathematics Subject Classification: 11E04; 14C25

Keywords and Phrases: Quadratic forms, Witt indices, Chow groups, correspondences.

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