Finitude uniforme pour les cycles de codimension 2 sur les corps de nombres

TL;DR

This paper establishes uniform bounds on the torsion subgroup of the Chow group of codimension 2 cycles for smooth projective varieties over number fields, extending previous finiteness results to a family setting.

Contribution

It provides the first uniform bounds for the torsion subgroup of $CH^2$ in families of varieties over number fields, under the condition $H^2(X, \\mathcal{O}_X)=0$.

Findings

Proves finiteness of $CH^2(X)_{tors}$ under certain conditions.

Provides explicit uniform bounds for the torsion subgroup across families.

Extends previous finiteness results to a uniform, family-wide context.

Abstract

Soit $X$ une vari\'et\'e projective et lisse, d\'efinie sur un corps de nombres. Sous l'hypoth\`ese $H^2(X,\mathcal O_X)=0,$ Colliot-Th\'el\`ene et Raskind ont d\'emontr\'e que le sous-groupe de torsion $CH^2(X)_{tors}$ du groupe de Chow en codimension $2$ est fini. Dans cette note, on donne des bornes uniformes pour le groupe fini $CH^2(X)_{tors}$ quand $X$ varie en famille. Let $X$ be a smooth projective variety defined over a number field. Assuming $H^2(X,\mathcal O_X)=0,$ Colliot-Th\'el\`ene and Raskind proved that the torsion subgroup $CH^2(X)_{tors}$ in the Chow group of cycles of codimension $2$ is finite. In this note, we give uniform bounds for the finite group $CH^2(X)_{tors}$ when $X$ varies in a family.