Submission: 2016, Feb 23, revised: 2017, Apr 20
We develop the version of the J-invariant for hermitian forms over quadratic extensions in a similar way Alexander Vishik did it for quadratic forms.This discrete invariant contains informations about rationality of algebraic cycles on the maximal unitary grassmannian associated with a hermitian form over a quadratic extension. The computation of the canonical $2$-dimension of this grassmannian in terms of the $J$-invariant is provided, as well as a complete motivic decomposition.
2010 Mathematics Subject Classification: 14C25 ; 11E39
Keywords and Phrases: Hermitian and quadratic forms, grassmannians, Chow groups and motives
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