The Néron–Severi group $N^{1}(X)$ of $X$ is the group ${\rm Pic}(X)\otimes_{\mathbb{Z}}{\mathbb{R}}$ modulo the subspace generated by the numerically trivial line bundles. For a closed $(1,1)$-form $\theta$, let $\{\theta\}$ be the associated de Rham class, given by $\{\theta\}=\theta_{\mathscr{X}}|_{X}\in N^{1}(X)$ for any algebraic ${K^{\circ}}$-model ${\mathscr{X}}$ on which $\theta$ is determined by $\theta_{\mathscr{X}}\in N^{1}({\mathscr{X}}/S)$. If $\theta$ is semipositive, then $\{\theta\}$ is nef.

To see this, we choose any closed curve $C$ in $X$ and non-zero $\rho$ in the maximal ideal of the valuation ring ${K^{\circ}}$. Then using the divisorial intersection theory in [Gub98], we have

Since $\theta_{\mathscr{X}}$ is nef, the degree of the special fibre $\overline{C}_{s}$ is non-negative proving the claim.