$V$ for every $j$. As usual, by perturbing $m_j$s suitably, we get a desired semi-ample $\mathbb Q$-divisor $D^\dag$ on $V$. For the details, see, for example, the proof of [@fujino-fundamental Lemma 4.15].

Next, we treat a very useful covering trick, which is essentially due to Yujiro Kawamata. We will use it in the proof of Theorem \[x-thm9.3\].

\[x-lem3.3\] Let $f:V\to W$ be a projective surjective morphism between smooth quasi-projective varieties and let $H$ be a Cartier divisor on $W$. Let $d$ be an arbitrary positive integer. Then we can take a finite flat morphism $\tau:W'\to W$ from a smooth quasi-projective variety $W'$ and a Cartier divisor $H'$ on $W'$ such that $\tau^*H\sim dH'$ and that $V'=V\times _W W'$ is a smooth quasi-projective variety with $\omega_{V'/W'}=\rho^*\omega_{V/W}$, where $\rho:V'\to V$. By construction, we may assume that $\tau:W'\to W$ is Galois.

We take general very ample Cartier divisors $D_1$ and $D_2$ with the following properties.

-   $H\sim D_1-D_2$,

-   $D_1$, $D_2$, $f^*D_1$, and $f^*D_2$ are smooth,

-   $D_1$ and $D_2$ have no common components, and

-   ${{\operatorname{Supp}}}(D_1+D_2)$ and ${{\operatorname{Supp}}}(f^*D_1+f^*D_2)$ are simple normal crossing divisors.

We take a finite flat cover due to Kawamata with respect to $W$ and $D_1+D_2$. Then we obtain $\tau:W'\to 