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 $$J^{i,p}(X/k)_{\rm tor} = J^{i,p}(X/\mathbf{C})_{\rm tor}\ ?$$

**Example: the case $i=2,p=1.$**

As suggested in the comment, one can think about the case $i = 2$, $j=1$ first. Here $\mathbf{Z}(1)_{\mathcal{D}} \simeq\mathbf{G}_{\rm m}[-1]$ and $H^2_{\mathcal{D}}(X,\mathbf{Z}(1)) = H^1(X,\mathbf{G}_{\rm m}) = \text{Pic}(X)$. In this case, we have an exact sequence:

$$H^1(X,\mathbf{G}_{\rm a}) \to \text{Pic}(X)\xrightarrow{c_1} H^2(X,\mathbf{Z}(1))$$

that identifies the extension

$$0\to J^{i,p}(X/\mathbf{C})\to H^i_{\mathcal{D}}(X,\mathbf{Z}(p))\to\text{Hdg}^{i,p}(X/\mathbf{C})\to 0$$

with

$$0\to \text{Pic}^0(X)\to\text{Pic}(X)\to\text{NS}(X)\to 0$$

whence $J^{2,1}(X/\mathbf{C}) = \text{Pic}^0(X)$. By GAGA we have $\text{Pic}^0(X)\simeq\text{Pic}^0(\mathcal{X}_{\mathbf{C}})$ and since $\mathbf{C}$ is separably closed $$\text{Pic}^0(\mathcal{X}_{\mathbf{C}}) = \underline{\text{Pic}}^0_{\mathcal{X}_{\mathbf{C}}/\mathbf{C}}(\mathbf{C}) = (\underline{\text{Pic}}^0_{\mathcal{X}/k}\times_k\mathbf{C})(\mathbf{C}).$$

If $k$ is separably closed too, then indeed the torsion subgroup of $\text{Pic}^0(X)$ agrees with that of $\underline{\text{Pic}}^0_{\mathcal{X}/k}(k)$, since the kernel of multiplication by $n$ on $\underline{\text{Pic}}_{\mathcal{X}/k}^0$ is an étale group scheme for every $n$ (since $k$ of characteristic zero).

Since $\underline{\text{Pic}}^0_{\mathcal{X}/k}(k) = J^{2,1}(X/k)$, we indeed get

$$J^{2,1}(X/\mathbf{C})_{\rm tor} = J^{2,1}(X/k)_{\rm tor}.$$