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On the cycle map for torsion algebraic cycles of codimension two. (English) Zbl 0764.14004
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.

##### MSC:
 14C25 Algebraic cycles 14C05 Parametrization (Chow and Hilbert schemes)
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##### References:
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