On standard norm varieties

Karpenko, Nikita A.; Merkurjev, Alexander S.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 177-216.

Pour un nombre premier $p$ et un corps $F$ de caractéristique $0$, soit $X$ la variété de norme d’un symbole dans le groupe de cohomologie galoisienne $H^{n+1}(F,{\mu }_p^{\otimes n})$ (avec $n\ge 1$) construite au cours de la démonstration de la conjecture de Bloch-Kato. Le résultat principal de cet article affirme que le corps des fonctions $F\left(X\right)$ a la propriété suivante  : pour toute variété équidimensionnelle $Y$, l’homomorphisme de changement de corps $\mathrm{CH}\left(Y\right)\to \mathrm{CH}\left(Y_{F\left(X\right)}\right)$ de groupes de Chow à coefficients entiers localisés en $p$ est surjectif en codimension $<(dimX)/(p-1)$. Une des composantes principales de la preuve est le calcul de groupes de Chow du motif de Rost généralisé (un variant du résultat principal indépendant de ceci est proposé dans l’appendice). Un autre ingrédient important est la $A$-trivialité de $X$, la propriété qui dit que pour toute extension de corps $L/F$ avec $X\left(L\right)\ne \varnothing$, l’homomorphisme de degré pour ${\mathrm{CH}}_0\left(X_L\right)$ est injectif. La preuve fait apparaître la théorie de correspondances rationnelles revue dans l’appendice.

Let $p$ be a prime integer and $F$ a field of characteristic $0$. Let $X$ be the norm variety of a symbol in the Galois cohomology group $H^{n+1}(F,{\mu }_p^{\otimes n})$ (for some $n\ge 1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F\left(X\right)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\mathrm{CH}\left(Y\right)\to \mathrm{CH}\left(Y_{F\left(X\right)}\right)$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $<(dimX)/(p-1)$. One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in the appendix). Another important ingredient is $A$-triviality of $X$, the property saying that the degree homomorphism on ${\mathrm{CH}}_0\left(X_L\right)$ is injective for any field extension $L/F$ with $X\left(L\right)\ne \varnothing$. The proof involves the theory of rational correspondences reviewed in the appendix.

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