Kimura-finiteness of quadric fibrations over smooth curves

TL;DR

This paper proves that the mixed motive of a quadric fibration over a smooth curve is Kimura-finite by utilizing recent advances in noncommutative mixed motives, contributing to the understanding of motives in algebraic geometry.

Contribution

It introduces a proof of Kimura-finiteness for motives of quadric fibrations over smooth curves using noncommutative mixed motives, a novel approach in the field.

Findings

The mixed motive of a quadric fibration over a smooth curve is Kimura-finite.

Application of noncommutative mixed motives to classical problems in algebraic geometry.

Establishes a new link between noncommutative motives and classical motivic finiteness properties.

Abstract

In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.