Pure motives with representable Chow groups

TL;DR

This paper proves that Chow motives over an algebraically closed field with all Chow groups representable are generated by twisted motives of curves, using the theory of birational motives by Kahn and Sujatha.

Contribution

It establishes a classification of Chow motives with representable Chow groups as being generated by twisted motives of curves, expanding understanding of motive structures.

Findings

Chow motives with all Chow groups representable belong to a specific subcategory.

The subcategory is generated by twisted motives of curves.

The proof uses Kahn and Sujatha's theory of birational motives.

Abstract

Let $k$ be an algebraically closed field. We show using Kahn's and Sujatha's theory of birational motives that a Chow motive over $k$ whose Chow groups are all representable belongs to the full and thick subcategory of motives generated by the twisted motives of curves. -- Motifs purs dont les groupes de Chow sont repr\'esentables. Soit $k$ un corps alg\'ebriquement clos. Nous prouvons, en nous servant de la th\'eorie des motifs birationnels d\'evelopp\'ee par Kahn et Sujatha, qu'un motif de Chow d\'efini sur $k$ dont les groupes de Chow sont tous repr\'esentables appartient \`a la sous-cat\'egorie pleine et \'epaisse des motifs engendr\'ee par les motifs de courbes tordus.