We define $\mathcal{Z}^{1,1}(X)_{\mathbb{Q}}$ as the direct limit

where ${{\mathscr{X}}}$ runs over the isomorphism classes of models of $X$. The space of closed $(1,1)$-forms on $X$ is defined as $\mathcal{Z}^{1,1}(X)\coloneqq\mathcal{Z}^{1,1}(X)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{R}$. Let $L$ be a line bundle on $X$. Let ${\|\ \|}$ be a model metric on $L^{\mathrm{an}}$ which is determined on ${{\mathscr{X}}}$ by a model ${\mathscr{L}}$ of $L^{\otimes m}$. We multiply the class of ${\mathscr{L}}$ in $N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ by $m^{-1}$ which determines a well defined class $c_{1}(L,{\|\ \|})\in\mathcal{Z}^{1,1}(X)_{\mathbb{Q}}\subseteq\mathcal{Z}^{1,1}(X)$ called the curvature form $c_{1}(L,{\|\ \|})$ of $(L,{\|\ \|})$.

A closed $(1,1)$-form $\theta$ is called semipositive if it is represented by a nef element $\theta_{{\mathscr{X}}}\in N^{1}({{\mathscr{X}}}/S)$ for some model ${{\mathscr{X}}}$ of $X$. We say that a model metric ${\|\ \|}$ on $L^{\mathrm{an}}$ for a line bundle $L$ on $X$ is semipositive if the same holds for the curvature form $c_{1}(L,{\|\ \|})$.