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If $X$ is a proper smooth complex analytic space, one can define Chow groups of analytic cycles on $X$ the usual way.

We have a cycle map

$$c^p_X: \text{CH}^p(X) \to \text{H}^{2p}_{D}(X,\mathbf{Z}(p))$$ to Deligne cohomology of $X$.

Is $c^p_X$ an isomorphism? Is $c^p_X\otimes\mathbf{Q}$ an isomorphism?

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Unless I misunderstand your definition, wouldn't $CH^p(X)$ coincide with the usual Chow group when $X$ is smooth projective, by GAGA? And of course Deligne cohomology would be the same. So the answers should be no and no, for a general variety $X$.

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  • $\begingroup$ Thanks for your answer. Do you have an example of a smooth projective variety for which $c^p_X$ is not surjective? I have some examples for non injective $c^p_X$. In other words, surjectivity of $c^p_X\otimes\mathbf{Q}$ is conjectured only onto $\text{Hdg}^{p,p}(X)\otimes\mathbf{Q}$, but not onto the full $H^{2p}_D(X,\mathbf{Z}(p))$. Correct? $\endgroup$
    – user113452
    Commented Feb 10, 2018 at 6:16
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    $\begingroup$ $c^p_X$ and $c^p_X\otimes \mathbb{Q}$ are usually not surjective (for $p>1$). Indeed, if $c^p_X$ were surjective, its restriction to the group $CH^p(X)_{hom}$ of cohomologically trivial cycles would surject onto the intermediate Jacobian $J^p(X)$. But for a general hypersurface of sufficiently high degree in $\mathbb{P}^{2p}$, Griffiths proved long time ago that the image of $c^p_X$ is countable. $\endgroup$
    – abx
    Commented Feb 10, 2018 at 17:31

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