If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of $\omega$.

My question is what's the analog for degenerated Kähler metrics? For example, if $\omega$ belongs to a semi-ample class, we know that $\omega$ must degenerate along some exceptional sub-varieties. What's the Chern-Weil theory for this case? Any reference for this? Thanks.