Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology.

What properties does the subgroup of torsion elements of $H^i_{\mathcal{D}}(X,\mathbf{Z}(p)$ have? 

- For instance, is it finite? Does it contain divisible elements?

Since $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ is an extension of a finitely generated $\mathbf{Z}$-module by a quotient of a graded in the Hodge filtration on de Rham cohomology of $X$, one expects the answer depends on $i,p$.

More precisely, the question might as well be asked about the quotient group

$$J^{i,p}(X/\mathbf{C}) := \frac{H^i_{\rm dR}(X/\mathbf{C})}{F^pH^i_{\rm dR}(X/\mathbf{C}) + H^i(X,\mathbf{Z}(p))}$$

- What if one restricts attention to the set $J^{i,p}(X/k)$ of those classes in $J^{i,p}(X/\mathbf{C})$ that come from algebraic cycles on the algebraization of $X$, and are defined on a fixed subfield $k\subset\mathbf{C}$? 

More precisely, if $\mathcal{X}$ is a smooth projective algebraic $k$-variety such that $(\mathcal{X}\otimes_k\mathbf{C})^{\rm an}\simeq X$, and $c : H^i_{\mathcal{M}}(\mathcal{X},\mathbf{Z}(p))\to H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ is the cycle map, define $J^{i,p}(X/k)$ to be

$$J^{i,p}(X/k) := J^{i,p}(X/\mathbf{C})\times_{H^i_{\mathcal{D}}(X,\mathbf{Z}(p))}H^i_{\mathcal{M}}(\mathcal{X},\mathbf{Z}(p))$$


What can be said about the torsion subgroup of $J^{i,p}(X/k)$?

If $k$ is algebraically closed, do we have
$$J^{i,p}(X/k)_{\rm tor} = J^{i,p}(X/\mathbf{C})_{\rm tor}\ ?$$