Jacques Tits' motivic measure

TL;DR

This paper introduces the Tits' motivic measure using noncommutative motives and applies it to classify algebraic varieties like Severi-Brauer varieties and quadrics based on their Grothendieck classes, revealing new isomorphism and birationality criteria.

Contribution

It constructs a new motivic measure and demonstrates its effectiveness in classifying certain algebraic varieties via Grothendieck classes, extending previous results.

Findings

Severi-Brauer varieties of period 2 are isomorphic iff they have the same Grothendieck class.

Varieties of period 2 to 6 with the same Grothendieck class are birational.

Quadrics of degree 6 are isomorphic iff they share the same Grothendieck class.

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.