The cohomological Hodge conjecture for singular varieties

Let $X$ be a complex projective variety defined over $\overline{\mathbb Q}$. A weight $2p$ Hodge cycle is a class in $H^{2p}(X,\mathbb Q(p))$ of Hodge type $(p,p)$. Let

$$H_M^{2p}(X,\mathbb Q(p))$$

be motivic cohomology, and let the map to $H^{2p}(X,\mathbb Q(p))$ be the map constructed in the cited theorem. The cohomological Hodge conjecture. Every weight $2p$ Hodge cycle in $H^{2p}(X,\mathbb Q(p))$ lies in the image of

$$H_M^{2p}(X,\mathbb Q(p))\longrightarrow H^{2p}(X,\mathbb Q(p)).$$

This extends the Lefschetz $(1,1)$ theorem from projective varieties to higher codimension and singular varieties. The source notes that the condition that $X$ be defined over $\overline{\mathbb Q}$ is essential, and gives examples showing that dropping it can lead to counterexamples.