Jacques Tits' motivic measure

Abstract

Making use of the recent theory of noncommutative motives, we construct a new motivic measure, which we call the Tits' motivic measure. As a first application, we prove that two Severi-Brauer varieties (or more generally twisted Grassmannian varieties), associated to central simple algebras of period 2, have the same Grothendieck class if and only if they are isomorphic. As a second application, we show that if two Severi-Brauer varieties, associated to central simple algebras of period 2, 3, 4, 5 or 6, have the same Grothendieck class, then they are necessarily birational. As a third application, we prove that two quadric hypersurfaces (or more generally involution varieties), associated to quadratic forms of degree 6, have the same Grothendieck class if and only if they are isomorphic. This latter result also holds for products of such quadrics. Finally, as a fourth application, we show in certain cases that two products of conics have the same Grothendieck class if and only if they are isomorphic; this refines a result of Kollar.