  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 hermitan
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.
