. A closed positive $(1,1)$-current $T$ over a compact complex manifold $X$ possesses analytic singularities if there exists $k\in\mathbb{N}_{>0}$ such that;
a) the ideal sheaf ${\cal J}^{kT}$ is coherent,
b) if $\mu:\tilde{X}\rightarrow X$ a modification such that $\mu^{*}{\cal J}^{kT}={\cal O}(-D)$, with $D$ an effective divisor with normal crossing, then $\mu^{*}T=\lambda[D]+\alpha$, with $\lambda\in\mathbb{R}_{>0}$ and $\alpha$ smooth.