and then denote by $\geq$ the partial order relation on ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ induced by the convex cone ${\mathcal{E}}$.

The Néron-Severi space
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Given a line bundle $L$ on $X$, each smooth Hermitian metric $h$ on $L$ locally writes as $h(x,v)=|v|^2e^{-2\varphi(x)}$ for some smooth local weight $\varphi$; the curvature form $\Theta_h(L):=\frac{i}{\pi}{\partial\overline{\partial}}\varphi$ is a globally defined real $(1,1)$-form, whose class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ we denote by $c_1(L)$, the first Chern class of $L$. We write $dd^c=\frac{i}{\pi}{\partial\overline{\partial}}$ for short. A singular Hermitian metric $h$ on $L$ is by definition a metric $h=h_{\infty}e^{-2\varphi}$, where $h_{\infty}$ is a smooth Hermitian metric on $L$ and the weight $\varphi$ is a locally integrable function. The curvature current of $h$ is defined as $\Theta_h(L):=\Theta_{h_{\infty}}(L)+dd^c\varphi$; it also lies in $c_1(L)$. Conversely, given a smooth Hermitian metric $h_{\infty}$ on $L$, any closed real $(1,1)$-current $T$ in $c_1(L)$ can be written (by definition) as $T=\Theta_{h_{\infty}}(L)+dd^c\varphi$. But $\varphi$ is just a degree $0$ current $a$ $priori$. However, $\varphi$ is automatically $L^1_{loc}$ in case $T$ is almost positive (cf. section 2.2.2), thus each almost positive current $T$ in $c_1(L)$ is the curvature form of a singular Hermitian metric on $L$.\
The image of the homomorphism Pic$(X)\to{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ $L\mapsto c_1(L)$ is called the Néron-Severi group, denoted by 