Let $(L,h)$ be a pseudoeffective line bundle on $X$ equipped with a smooth metric $h$. Set $\omega:=\Theta_{h}$. Then

$$h_{min}:=h+V_{X,\omega}^{*}$$

is a positive singular metric of $L$ on $X$ with ”minimal singularities”. More precisely if $\psi$ is a positive singular metric of $L$ on $X$, then there exists a constant $C_{\psi}$ such that $\psi\leq h_{min}+C_{\psi}$.