O_{X^{(s)}_0}(L^{(s)}_0)&\subset f^s_*\mathcal O_{X^s_0}
\left(\sum_{i=1}^s p^*_i L_0\right)\\
&\simeq \bigotimes _{i=1}^s f_*\mathcal O_{X_0}(L_0), 
\end{split}$$ where $f^{(s)}=f^s\circ \rho:X^{(s)}_0\to Y_0$. By assumption, $L_0-kK_{X_0/Y_0}\sim_{\mathbb R} k\Delta_0$. Therefore, $$L^{(s)}_0\sim _{\mathbb R} k\left(K_{X^{(s)}_0/Y_0}
+\sum _{i=1}^s \pi^*_i \Delta_0\right)$$ by . We can take a nonempty Zariski open set $U$ of $Y_0$ such that $f$ is smooth over $U$, ${{\operatorname{Supp}}}\Delta$ is relatively simple normal crossing over $U$, and $\rho$ is an isomorphism over $U$. Then we see that $\left(X^{(s)}_0, \sum _{i=1}^s \pi^*_i \Delta_0\right)$ is log canonical over $U$.

Weakly positive sheaves and big sheaves {#x-sec4}
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We quickly see some basic properties of Viehweg’s weakly positive sheaves and big sheaves. For the details, see [@fujino-iitaka Chapter 3], [@viehweg1], [@viehweg2], and [@viehweg3].

\[x-def4.1\] Let $\mathcal F$ be a torsion-free coherent sheaf on a smooth quasi-projective variety $W$. We say that $\mathcal F$ is [*[weakly positive]{}*]{} if, for every positive integer $\alpha$ and every ample invertible sheaf $\mathcal H$, there exists a positive integer $\beta$ such that $\widehat{S}^{\alpha\beta}(\mathcal F)\otimes \mathcal H^{\otimes \beta}$ is generically generated by global sections. We say that a nonzero torsion-free coherent sheaf $\mathcal F$ is [*[big]{}*]{} (in the sense of Viehweg) if, for every ample invertible sheaf $\mathcal H$, there exists a positive integer $a$ such that 