On the cycle map for torsion algebraic cycles of codimension two
For complete varieties \(X\) over a field \(k\) and positive integers \(n\), conditions that entail the injectivity of the map \(\rho_{n,tor}:CH^ 2(X)_{tor}\to H^ 4(X,\mu_ n^{\otimes 2})\) obtained by composing the inclusion \(CH^ 2(X)_{tor}\hookrightarrow CH^ 2(X)\), the quotient map \(CH^ 2(X)\to CH^ 2(X)/n\) and the cycle map \({\rho_ n:CH^ 2(X)/n\to H^ 4(X,\mu_ n^{\otimes 2})}\) are discussed \((\mu_ n\) is the sheaf of \(n\)-th roots of unity on \(X\) and \(CH^ 2(X)\) is the group of rational classes of codimension 2 cycles on \(X)\). We cannot reproduce here the several sets of subtle conditions that are proved to suffice for the injectivity, but the flavour of them can be appreciated by the following sample case: if \(X\) is projective and smooth over a number field \(k\), if the Picard variety of \(X\) has potentially good reduction, and if \(H^ 2(X_{zar},{\mathcal O}_ X)=0\), then there exists an integer \(N>0\) such that \(\rho_{n,tor}\) is injective for any \(n\) divisible by \(N\). Let us also state one of the theorems that are established with the methods introduced to prove the injectivity results: If \(X\) is smooth over a field \(k\) which is a finitely generated extension of \(\mathbb{Q}\), if \(H^ 2(X_{zar}{\mathcal O}_ X)=H^ 1(X_{zar},{\mathcal O}_ X)=0\), and if \(X\) has a \(k\)-rational point, then \(CH^ 2(X)_{tor}\) is finite.
scientific article; zbMATH DE number 59783
scientific article