We set ${\rm Pic}({\mathscr{X}})_{\mathbb{R}}\coloneqq{\rm Pic}({\mathscr{X}})\otimes_{\mathbb{Z}}{\mathbb{R}}$. We define the Néron-Severi group as the ${\mathbb{R}}$-vector space ${\rm Pic}({\mathscr{X}})_{\mathbb{R}}$ modulo the subspace generated by numerically trivial line bundles. We denote this space by $N^{1}({\mathscr{X}}/S)$, where $S:={\rm Spec}({K^{\circ}})$. The space of closed $(1,1)$-forms on $X$ is defined as the direct limit

$$\mathcal{Z}^{1,1}(X)\coloneqq\varinjlim N^{1}({\mathscr{X}}/S)$$

where the limit is taken over all algebraic ${K^{\circ}}$-models of $X$. We say that a closed $(1,1)$-form $\theta$ is determined on some model ${\mathscr{X}}$ if it is in the image of the map $N^{1}({\mathscr{X}}/S)\to\mathcal{Z}^{1,1}(X)$. The canonical map $N^{1}({\mathscr{X}}/S)\to\mathcal{Z}^{1,1}(X)$ induces a map $dd^{c}:\mathcal{D}(X)\to\mathcal{Z}^{1,1}(X)$.