A smooth Kähler metric $\Omega$ on $V$ is a $\sim$-equivalence class of smooth Kähler potentials. A semi-Kähler (resp. Kähler) current on $V$ is a $\sim$-equivalence class of semi-Kähler (resp. Kähler) potentials.

A semi-Kähler current $\Omega=(U_{i},\varphi_{i})_{i\in I}\mod\sim$ is said to have $L_{loc}^{\infty}$ (resp. ${\mathcal{C}}^{0}$, resp. Hölder continuous) potentials iff each $\varphi_{i}$ is $L_{loc}^{\infty}$ (resp. ${\mathcal{C}}^{0}$, resp. Hölder continuous).