Jacques Tits motivic measure

Gonçalo Tabuada

doi:10.1007/s00208-021-02292-6

Abstract

In this article we construct a new motivic measure called the Jacques Tits motivic measure. As a first main application, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to 2-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period {3, 4, 5, 6}, have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension 6 or to quadratic forms of arbitrary dimension defined over a base field k with I^3(k)=0, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.

Highlights

Let k be a field and Var(k) the category of varieties, i.e., reduced separated k-schemes of finite type
Grothendieck to Serre, is defined as the quotient of the free abelian group on the set of isomorphism classes of varieties [X ] by the “scissor” relations [X ] = [Y ] + [X \Y ], where Y is a closed subvariety of X
The multiplication law is induced by the product of varieties