## Subtle Characteristic Classes and Hermitian Forms

##### Doc. Math. 24, 2493-2523 (2019)
DOI: 10.25537/dm.2019v24.2493-2523

### Summary

Following \textit{A. Smirnov} and \textit{A. Vishik} [Subtle characteristic classes'', Preprint, \url{arXiv:1401.6661v1}], 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 Čech 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 Čech 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.

### Mathematics Subject Classification

11E39, 14F42, 20G15, 55R40

### Keywords/Phrases

Hermitian forms, motivic cohomology, Nisnevich classifying space, characteristic classes

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### Affiliation

Tanania, Fabio
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK