# Subtle characteristic classes and Hermitian forms

## TL;DR

This paper computes the motivic cohomology of classifying spaces for unitary groups linked to hermitian forms, introducing subtle characteristic classes that relate to existing classes in orthogonal cases and describing the motive of associated torsors.

## Contribution

It introduces subtle characteristic classes for hermitian forms in motivic cohomology and relates them to orthogonal classes, providing new insights into the motives of hermitian form torsors.

## Key findings

- Computed the motivic cohomology ring of the classifying space for unitary groups.
- Established relations between subtle characteristic classes and subtle Stiefel-Whitney classes.
- Described the motive of the torsor associated with a hermitian form.

## Abstract

Following [14], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the \v{C}ech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the \v{C}ech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfister form. Moreover, we present a description of the motive of the torsor corresponding to a hermitian form in terms of its subtle characteristic classes.