Grothendieck classes of quadric hypersurfaces and involution varieties

TL;DR

This paper links the Grothendieck classes of quadrics and involution varieties to their algebraic invariants, showing that equal classes imply isomorphism in many cases, thus connecting motive theory with algebraic geometry.

Contribution

It establishes a new connection between Grothendieck classes and algebraic invariants of quadrics and involution varieties, combining noncommutative and classical motive theories.

Findings

Equal Grothendieck classes imply identical even Clifford algebras.

In many cases, equal classes mean the varieties are isomorphic.

The results apply over local and global fields.

Abstract

In this article, by combining the recent theory of noncommutative motives with the classical theory of motives, we prove that if two quadrics (or, more generally, two involution varieties) have the same Grothendieck class, then they have the same even Clifford algebra and the same signature. As an application, we show in numerous cases (e.g., when the base field is a local or global field) that two quadrics (or, more generally, two involution varieties) have the same Grothendieck class if and only if they are isomorphic.