If we start with $\omega_{o}$ Kähler, and the log-resolution is non trivial, $\mu^{*}\omega_{o}$ is not Kähler anymore.

This method that dates back to [Ko] can be used to prove a variant of [Y], Theorem 7 p. 399 where the divisor of $s_{2}$ is a simple normal crossing divisor, under the sole assumption that $\int_{M}|s_{2}|^{-2k_{2}}<\infty$.

Now, it could not have been used to prove Theorem 8 p. 403 in 1978 since log-resolutions force the use of Monge-Ampère equations with degenerate L.H.S, for which the ${\mathcal{C}}^{0}$-estimate proved here was not available then.