Subtle Characteristic Classes and Hermitian Forms

Fabio Tanania

Documenta mathematica, Tome 24 (2019), pp. 2493-2523

Résumé

Following A. Smirnov and A. Vishik ["Subtle characteristic classes", Preprint, 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.

Détail

DOI : 10.4171/dm/732

Classification : 11E39, 14F42, 20G15, 55R40
Mots-clés : characteristic classes, motivic cohomology, Hermitian forms, Nisnevich classifying space