A model of $X$ is given by a proper flat scheme ${{\mathscr{X}}}$ over $S:={\rm Spec}\,K^{\circ}$ together with an isomorphism $h$ between $X$ and the generic fiber ${{\mathscr{X}}}_{\eta}$ of the $S$-scheme ${{\mathscr{X}}}$ which we read as an identification. Given a model ${{\mathscr{X}}}$ of $X$ there is a canonical surjective reduction map ${\mathrm{red}}\colon X^{\mathrm{an}}\longrightarrow{{\mathscr{X}}}_{s}$ where ${{\mathscr{X}}}_{s}$ denotes the special fiber ${{\mathscr{X}}}\otimes_{K^{\circ}}\tilde{K}$ of ${{\mathscr{X}}}$ over $S$.

Let $L$ be a line bundle on the proper variety $X$. A model of $(X,L)$ or briefly a model of $L$ is given by a model $({{\mathscr{X}}},h)$ of $X$ together with a line bundle ${\mathscr{L}}$ on ${{\mathscr{X}}}$ and an isomorphism $h^{\prime}$ between $L$ and $h^{*}({\mathscr{L}}|_{\mathscr{X}_{\eta}})$ which we read as an identification.

Given a model $({{\mathscr{X}}},{\mathscr{L}})$ of $(X,L^{\otimes m})$ for some $m\in\mathbb{N}_{>0}$ there is a unique metric ${\|\ \|}_{\mathscr{L}}$ on $L^{\mathrm{an}}$ over $X^{\mathrm{an}}$ which satisfies the following: Given an open subset ${\mathscr{U}}$ of ${{\mathscr{X}}}$, a frame $t$ of ${\mathscr{L}}$ over ${\mathscr{U}}$, and a section $s$ of $L$ over $U=X\cap{\mathscr{U}}$ we write $s^{\otimes m}=ht$ for some regular function $h$ on $U$ and get $\|s\|=\sqrt[m]{|h|}$ on $U^{\mathrm{an}}\cap{\mathrm{red}}^{-1}({\mathscr{U}}_{s})$. Such a metric on $L^{\mathrm{an}}$ is called a model metric determined on ${{\mathscr{X}}}$.

A model metric ${\|\ \|}$ on ${\mathcal{O}}_{X^{\mathrm{an}}}$ induces a continuous function $f=-\log\|1\|\colon X^{\mathrm{an}}\to\mathbb{R}$. The space of model functions

has a natural structure of a $\mathbb{Q}$-vector space. We say that a model function $f=-\log\|1\|$ is determined on a model ${{\mathscr{X}}}$ if the model metric ${\|\ \|}$ is determined on ${{\mathscr{X}}}$. A vertical divisor $D$ on ${{\mathscr{X}}}$ determines a model ${\mathcal{O}}(D)$ of ${\mathcal{O}}_{X}$ and an associated model function $\varphi_{D}\coloneqq-\log\|1\|_{{\mathcal{O}}(D)}$. Such model functions are called $\mathbb{Z}$-model functions. Let ${\mathfrak{a}}$ denote a vertical ideal of ${{\mathscr{X}}}$. Let $E$ denote the exeptional divisor of the blowup of ${{\mathscr{X}}}$ in ${\mathfrak{a}}$. Then $\log|{\mathfrak{a}}|:=\varphi_{E}$ is called the $\mathbb{Z}$-model function defined by the vertical ideal ${\mathfrak{a}}$.

Consider a model ${{\mathscr{X}}}$ of the proper variety $X$ over $K$. The rational vector space space $N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ is by definition the quotient of ${{\rm Pic}\,}({{\mathscr{X}}})_{\mathbb{Q}}\coloneqq{\rm Pic}({{\mathscr{X}}})\otimes_{\mathbb{Z}}\mathbb{Q}$ by the subspace generated by classes of line bundles ${\mathscr{L}}$ such that ${\mathscr{L}}\cdot C=0$ for each closed curve $C$ in the special fiber ${{\mathscr{X}}}_{s}$. Note that $N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ is finite dimensional by applying [Kle66, Prop. IV.1.4] to ${{\mathscr{X}}}_{s}$. We define $N^{1}({{\mathscr{X}}}/S)\coloneqq N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{R}$. An element $\alpha\in N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ (resp. $\alpha\in N^{1}({{\mathscr{X}}}/S))$ is called nef if $\alpha\cdot C\geq 0$ for all closed curves $C$ in ${{\mathscr{X}}}_{s}$. We call a line bundle ${\mathscr{L}}$ on ${{\mathscr{X}}}$ nef if the class of ${\mathscr{L}}$ in $N^{1}({{\mathscr{X}}}/S)$ is nef.

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 $c_{1}(L,{\|\ \|})$.

For $\theta\in\mathcal{Z}^{1,1}(X)$ we denote by

the set of $\theta$-plurisubharmonic ($\theta$-psh for short) model functions. Recall from [GM16, Prop. 3.12] that the set ${\rm PSH}_{\mathscr{D}}(X,\theta)$ is stable under the formation of max.

If $Y$ is a proper variety over an arbitrary field $k$, we denote by $N^{1}(Y)_{\mathbb{Q}}$ the rational vector space ${{\rm Pic}\,}(Y)\otimes\mathbb{Q}$ modulo numerical equivalence. Similarly, we denote by $N^{1}(Y)=N^{1}(Y)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{R}$ the real vector space ${{\rm Pic}\,}(Y)\otimes\mathbb{R}$ modulo numerical equivalence. A class in $N^{1}(Y)$ is called ample if it is an $\mathbb{R}_{>0}$-linear combination of classes induced by ample line bundles on $Y$. An element $\alpha\in N^{1}(Y)_{\mathbb{Q}}$ (resp. $\alpha\in N^{1}(Y))$ is called nef if $\alpha\cdot C\geq 0$ for all closed curves $C$ in $Y$.

The restriction maps $N^{1}({{\mathscr{X}}}/S)\rightarrow N^{1}(X),\,[{\mathscr{L}}]\mapsto[{\mathscr{L}}|_{X}]$ induce a linear map $\{\phantom{a}\}:\mathcal{Z}^{1,1}(X)\longrightarrow N^{1}(X),\,\theta\mapsto\{\theta\}$. We call $\{\theta\}$ the de Rham class of $\theta$.