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?



Analytic cycles on complex-analytic spaces
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 $X$.


