Galois descent on the second Chow group: development and applications (Q273859)

This paper is concerned with the study of the group \(\mathrm{CH}^2(X)\) of codimension \(2\) cycles on a smooth projective variety \(X\) over a field of characteristic \(0\) and its behaviour under arbitrary field extensions. The results are especially interesting for geometrically rationally connected varieties, and in particular cubic hypersurfaces in \(\mathbb{P}^n\) for \(n\geq 4\).NEWLINENEWLINELet \(F\) be a field of characteristic \(0\) with a fixed algebraic closure \(\overline{F}\) and corresponding absolute Galois group \(G=\mathrm{Gal}(\overline{F}/F)\). Let \(X/F\) be a smooth projective geometrically connected variety. Write \(\overline{X}=X_{\overline{F}}\). The group \(\mathrm{CH}^2(X)\) of codimension \(2\) cycles under rational equivalence behaves in a complicated way with respect to field extensions; in particular, the map NEWLINE\[NEWLINE \psi:\mathrm{CH}^2(X)\rightarrow \mathrm{CH}^2(\overline{X})^G NEWLINE\]NEWLINE is neither injective nor surjective in general. The results of the paper gives some control over the kernel and the cokernel of this map in terms of a long exact sequence involving various cohomological invariants of \(X\), under increasingly stringent hypotheses on \(X\).NEWLINENEWLINEThere are many variants throughout the paper; let us quote two of them, in a somewhat less precise form than in the paper. To properly state them, we recall some notations. Let \(Y\) be one of \(X\) or \(\overline{X}\). We write \(\mathcal{K}_2\) for the Zariski sheaf on \(Y\) attached to the presheaf \(U\mapsto K_2(H^0(U,\mathcal{O}_Y))\) where \(K_2\) denotes Quillen \(K\)-theory. We denote by \(H^i_{\mathrm{nr}}(Y,\mathbb{Q}/\mathbb{Z}(2))\) the \(i\)-th unramified étale cohomology group on \(Y\) with coefficients in \(\mathbb{Q}/\mathbb{Z}(2)\), i.e., NEWLINE\[NEWLINE H^i_{\mathrm{nr}}(Y,\mathbb{Q}/\mathbb{Z}(2)):=\mathrm{Ker}[ H^i(k(Y),\mathbb{Q}/\mathbb{Z}(2))\rightarrow \sum_{y\in Y^{(1)}}H^i(k(y),\mathbb{Q}/\mathbb{Z}(1))] NEWLINE\]NEWLINE where the maps are residue maps. The first result we want to quote is: if \(\mathrm{Pic}(\overline{X})\) is torsion-free and \(X(F)\neq \emptyset\), then there is a long exact sequence NEWLINE\[NEWLINE\begin{multlined} 0\rightarrow \mathrm{Ker}(\psi)\rightarrow H^1(G,H^1(\overline{X},\mathcal{K}_2)) \rightarrow \mathrm{Ker}[H^3_{\mathrm{nr}}(X,\mathbb{Q}/\mathbb{Z}(2))/H^3_{\mathrm{nr}}(F,\mathbb{Q}/\mathbb{Z}(2))\\ \rightarrow H^3_{\mathrm{nr}}(\overline{X},\mathbb{Q}/\mathbb{Z}(2))]\rightarrow \mathrm{Coker}(\psi)\rightarrow H^2(G,H^1(\overline{X},\mathcal{K}_2)). \end{multlined}NEWLINE\]NEWLINE The second result we want to quote is: suppose \(X\) is a smooth cubic hypersurface in \(\mathbb{P}^n\) for \(n\geq 5\), and \(F\) contains \(\mathbb{C}\). Then the map \(\psi\) is surjective. We now say a word about the proofs. Writing again \(Y=X\) or \(\overline{X}\), we have a fundamental exact sequence NEWLINE\[NEWLINE 0\rightarrow \mathrm{CH}^2(Y)\rightarrow H^2(Y_{\mathrm{et}},\mathbb{Z}(2)) \rightarrow H^3_{\mathrm{nr}}(Y,\mathbb{Q}/\mathbb{Z}(2))\rightarrow 0 NEWLINE\]NEWLINE which, as many comparison statements between motivic cohomology and étale motivic cohomology, is deduced from the Beilinson-Lichtenbaum conjecture (in this case, the Merkurjev-Suslin theorem). Comparing this sequence for \(Y=X\) and \(\overline{X}\) through the Hochschild-Serre spectral sequence for étale motivic cohomology of \(\mathbb{Z}(2)\) and translating the hypothesis of divisibility of \(\mathrm{Pic}(\overline{X})\) in terms of the divisibility of \(H^0(\overline{X},\mathcal{K}_2)\) yields the proof of the first long exact sequence above.NEWLINENEWLINETo get more precise results for rationally connected varieties and in particular for cubic hypersurfaces, many additional arguments are required. Without getting into too much details, one needs to first control \(H^1(\overline{X},\mathcal{K}_2)\) in terms of \(\mathrm{Pic}(\overline{X})\otimes F^\times\), using a theorem of Raskind and the author. The next step consists in exploiting a ``decomposition of the diagonal'' hypothesis (which the author interprets as a ``small motive'' hypothesis) and the classical results of Bloch-Srinivas on the Hodge numbers, Picard and Brauer groups of such varieties. Finally, the finest statements (for cubic hypersurfaces in \(\mathbb{P}^5\)) require recent work of Claire Voisin on the integral Hodge conjecture in degree \(4\) for such hypersurfaces.