Critical varieties and motivic equivalence for algebras with involution

TL;DR

This paper extends the concept of motivic equivalence to all classical groups and some exceptional groups, introducing critical varieties and providing criteria for involutions to be isomorphic based on motivic data.

Contribution

It generalizes motivic equivalence characterization to broader classes of algebraic groups and involutions, introducing critical varieties and scalar extension criteria.

Findings

Motivic equivalence can be checked after scalar extension to an index reduction field.

Conditions are identified under which motivic equivalent involutions are isomorphic.

The work extends previous results from quadratic forms to all classical and some exceptional groups.

Abstract

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F_2, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.