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Note as well that along the Milnor diagonal $n=i$ there is evidence that the Beilinson-Hodge conjecture might be true even outside of the situation that $X$ is defined over a number field. See [AK09], [AS07], [Sai04] and [Sai09] for examples and further discussion.

For an example of $\mathrm{cl}_{i,n}$ failing to be surjective for $n\geq 1$, consider a K3 surface $X$ over $\mathbb{C}$ with points $P,Q$ such that $[P]\neq [Q]\in\mathrm{CH}^{2}_{\mathrm{deg}=0}(X)$. Let $U:=X\backslash\{P,Q\}$. Then $\mathrm{cl}_{2,1}:H^{3}_{\mathrm{Mot}}(U,\mathbb{Q}(2))\rightarrow H^{3}(U,\mathbb{Q}(2))$ is the zero map. Notice that $U$ is not defined over a number field here by the assumption, because Beilinson-Bloch predicts that $\mathrm{deg}:\mathrm{CH}^{2}(X)\xrightarrow{\simeq}\mathbb{Z}$ for $X$ defined over a number field (and it is known that $\mathrm{dim}_{\mathbb{Q}}\mathrm{CH}^{2}(X)_{\mathbb{Q}}=\infty$ for $X/\mathbb{C}$).

Note as well that along the Milnor diagonal $n=i$ there is evidence that the Beilinson-Hodge conjecture might be true even outside of the situation that $X$ is defined over a number field. See [AK09], [AS07], [Sai04] and [Sai09] for examples and further discussion.

In fact, $\mathrm{cl}_{i,n}$ factors through the cycle class map $\mathrm{cl}^{\mathcal{H}}_{i,n}$ into absolute Hodge cohomology:
\begin{equation*} \require{AMScd} \begin{CD} H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i)) @>\mathrm{cl}^{\mathcal{H}}_{i,n}>> H^{2i-n}_{\mathcal{H}}(X,\mathbb{Q}(i))\\ @. {_{\rlap{\ \mathrm{cl}_{i,n}}}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVV\\ @. \mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n}(X,\mathbb{Q}(i))) \end{CD} \end{equation*} and the vertical arrow is a surjection (the kernel is $\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i)))$). So you are really asking about the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$.

In fact, $\mathrm{cl}_{i,n}$ factors through the cycle class map $\mathrm{cl}^{\mathcal{H}}_{i,n}$ into absolute Hodge cohomology:
\begin{equation*} \require{AMScd} \begin{CD} H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i)) @>\mathrm{cl}^{\mathcal{H}}_{i,n}>> H^{2i-n}_{\mathcal{H}}(X,\mathbb{Q}(i))\\ @. {_{\rlap{\ \mathrm{cl}_{i,n}}}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVV\\ @. \mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n}(X,\mathbb{Q}(i))) \end{CD} \end{equation*} and the vertical arrow is a surjection (the kernel is the intermediate Jacobian $\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i)))$). So you are really asking about the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$. There the Beilinson-Hodge conjecture says that the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$ is dense, i.e. $\mathrm{cl}_{i,n}$ is surjective and the induced Abel-Jacobi map \begin{equation*} \mathrm{AJ}_{i,n}:H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i))_{\mathrm{hom}}\rightarrow\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i))) \end{equation*} has dense image. For the reasons given above, if $n\neq 0,i$ then this conjecture is for $X$ defined over a number field.

In fact, $\mathrm{cl}_{i,n}$ factors through the cycle class map $\mathrm{cl}^{\mathcal{H}}_{i,n}$ into absolute Hodge cohomology:
\begin{equation*} \require{AMScd} \begin{CD} H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i)) @>\mathrm{cl}^{\mathcal{H}}_{i,n}>> H^{2i-n}_{\mathcal{H}}(X,\mathbb{Q}(i))\\ @. {_{\rlap{\ \mathrm{cl}_{i,n}}}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVV\\ @. \mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n}(X,\mathbb{Q}(i))) \end{CD} \end{equation*} and the vertical arrow is a surjection (the kernel is $\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i)))$). So you are really asking about the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$.

Originally it was conjectured that $\mathrm{cl}_{i,n}^{\mathcal{H}}$ is always surjective (the so-called Beilinson-Hodge conjecture, see [Bei86, Conjecture 6]). First note that the case $n=0$ of the Beilinson-Hodge conjecture is equivalent to the Hodge conjecture (clear in the projective case, some work for quasi-projective $X$). But Jannsen [Jan90, Corollary 9.11] found counterexamples in the case $n=1$ (and counterexamples exist for all $n\geq 1$). Indeed, the surjectivity of $\mathrm{cl}^{\mathcal{H}}_{i,n}$ is tied up with the injectivity of certain higher Abel-Jacobi maps (see [Jan90, Corollary 9.10] and [dJL13, Theorem 4.9] more generally), and we know from Mumford that Abel-Jacobi (even $\otimes\mathbb{Q}$) can drastically fail to be injective in codimension $\geq 2$. Note, however, that the Bloch-Beilinson conjectures say that the higher Abel-Jacobi maps $\otimes\mathbb{Q}$ should be injective when $X$ is defined over a number field, so one could imagine that the Beilinson-Hodge conjecture is true for $X$ defined over number fields, and indeed if the Bloch-Beilinson conjectures and the Hodge conjecture are true then $\mathrm{cl}^{\mathcal{H}}_{i,n}$ is indeed surjective when $X$ is defined over a number field (see [Sai09]).

Note as well that along the Milnor diagonal $n=i$ there is evidence that the Beilinson-Hodge conjecture might be true even outside of the situation that $X$ is defined over a number field. See [AK09], [AS07], [Sai04] and [Sai09] for examples and further discussion.

Originally it was conjectured that $\mathrm{cl}_{i,n}$ is always surjective (the so-called Beilinson-Hodge conjecture, see [Bei86, Conjecture 6]). First note that the case $n=0$ of the Beilinson-Hodge conjecture is equivalent to the Hodge conjecture (clear in the projective case, some work for quasi-projective $X$). But Jannsen [Jan90, Corollary 9.11] found counterexamples in the case $n=1$ (and counterexamples exist for all $n\geq 1$). Indeed, the surjectivity of $\mathrm{cl}_{i,n}$ is tied up with the injectivity of certain higher Abel-Jacobi maps (see [Jan90, Corollary 9.10] and [dJL13, Theorem 4.9] more generally), and we know from Mumford that Abel-Jacobi (even $\otimes\mathbb{Q}$) can drastically fail to be injective in codimension $\geq 2$. Note, however, that the Bloch-Beilinson conjectures say that the higher Abel-Jacobi maps $\otimes\mathbb{Q}$ should be injective when $X$ is defined over a number field, so one could imagine that the Beilinson-Hodge conjecture is true for $X$ defined over number fields, and indeed if the Bloch-Beilinson conjectures and the Hodge conjecture are true then $\mathrm{cl}_{i,n}$ is indeed surjective when $X$ is defined over a number field (see [Sai09]).

Note as well that along the Milnor diagonal $n=i$ there is evidence that the Beilinson-Hodge conjecture might be true even outside of the situation that $X$ is defined over a number field. See [AK09], [AS07], [Sai04] and [Sai09] for examples and further discussion.

In fact, $\mathrm{cl}_{i,n}$ factors through the cycle class map $\mathrm{cl}^{\mathcal{H}}_{i,n}$ into absolute Hodge cohomology:
\begin{equation*} \require{AMScd} \begin{CD} H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i)) @>\mathrm{cl}^{\mathcal{H}}_{i,n}>> H^{2i-n}_{\mathcal{H}}(X,\mathbb{Q}(i))\\ @. {_{\rlap{\ \mathrm{cl}_{i,n}}}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVV\\ @. \mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n}(X,\mathbb{Q}(i))) \end{CD} \end{equation*} and the vertical arrow is a surjection (the kernel is $\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i)))$). So you are really asking about the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$.