$J$-invariant of hermitian forms over quadratic extensions

TL;DR

This paper extends the concept of the $J$-invariant to hermitian forms over quadratic extensions, providing tools to analyze algebraic cycles, compute canonical dimensions, and decompose motives of associated grassmannians.

Contribution

It introduces a new $J$-invariant for hermitian forms over quadratic extensions, paralleling Vishik's work on quadratic forms, and applies it to motivic decompositions and dimension calculations.

Findings

Computed the canonical 2-dimension of the grassmannian

Provided a complete motivic decomposition

Linked the $J$-invariant to rationality of algebraic cycles

Abstract

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.