Let $\mathcal{X}$ be a model of $X$. The space $N^{1}(\mathcal{X}/S)$ of (relative, codimension $1$) numerical equivalence classes on $\mathcal{X}$ is defined as the quotient of $\Pic(\mathcal{X})_{\mathbf{R}}$ by the subspace spanned by numerically trivial line bundles, i.e. those $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{R}}$ such that $\mathcal{L}\cdot C=0$ for all projective curves contained in a fiber of $\mathcal{X}\to S$. It is in fact enough to consider vertical curves, i.e. those contained in the special fiber $\mathcal{X}_{0}$. A class $\theta\in N^{1}(\mathcal{X}/S)$ is nef if $\theta\cdot C\geq 0$ for all such curves $C$.

We say that a closed $(1,1)$-form $\theta\in\mathcal{Z}^{1,1}(X)$ is determined on a given model $\mathcal{X}$ if it is the image of an element $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$. By definition, two classes $\theta\in N^{1}(\mathcal{X}/S)$ and $\theta^{\prime}\in N^{1}(\mathcal{X}^{\prime}/S)$ define the same element in $\mathcal{Z}^{1,1}(X)$ iff they pull back to the same class on a model dominating both $\mathcal{X}$ and $\mathcal{X}^{\prime}$.

The natural map $N^{1}(\mathcal{X}/S)\to N^{1}(X)$ gives rise to a map $\mathcal{Z}^{1,1}(X)\to N^{1}(X)$ which in fact is surjective. We refer to $\{\theta\}$ as the de Rham class of the closed $(1,1)$-form $\theta$. When $\theta$ is semipositive, the de Rham class $\{\theta\}\in N^{1}(X)$ is nef on $X$. In what follows, we shall mainly work with forms having ample de Rham class.

Any model function $f\in\mathcal{D}(X)$ induces a form $dd^{c}f\in\mathcal{Z}^{1,1}(X)$ as follows: for any determination $\mathcal{X}$ of $f$, $dd^{c}f$ is the class of the divisor $\sum_{i\in I}b_{i}f(x_{i})E_{i}$, where $\mathcal{X}_{0}=\sum_{i}b_{i}E_{i}$ and $x_{i}\in X$ is the divisorial point associated to $E_{i}$.