is birational, the trace map $\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\to \omega_V$ is a generically isomorphic injection $$\label{x-3.4}
\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\hookrightarrow \omega_V.$$ Since $\nu$ is finite, $$\label{x-3.5}
\nu^*\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\to \mathcal O_{V^\nu}(K_{V^\nu})$$ is surjective and the kernel of is the torsion part of $\nu^*\nu_*\mathcal O_{V^\nu}(K_{V^\nu})$. Therefore, by , we get an inclusion $$\label{x-3.6}
\mathcal O_{V^\nu}(K_{V^\nu})\hookrightarrow \nu^*\omega_V.$$ Let $n$ be a positive integer with $n\geq 2$. Then we have $$\mathcal O_{V^\nu}(nK_{V^\nu})=\mathcal O_{V^\nu}
(K_{V^\nu}+(n-1)K_{V^\nu})\hookrightarrow 
\mathcal O_{V^\nu}(K_{V^\nu})\otimes \nu^*\omega^{\otimes n-1}_V$$ by . Therefore, by taking $\nu_*$, we get $$\nu_*\mathcal O_{V^\nu}(nK_{V^\nu})
\hookrightarrow \nu_*\mathcal O_{V^\nu}(K_{V^\nu})\otimes 
\omega^{\otimes n-1}_V\hookrightarrow \omega^{\otimes n}_V$$ by . This is what we wanted.

By the above construction of and , it is obvious that the inclusion $$(\nu\circ \delta)_*\mathcal O_{V'}(nK_{V'}+E)\subset 
\omega^{\otimes n}_V$$ is an isomorphism over $U$.

\[x-lem3.6\] Let $f:X_0\to Y_0$ be a projective surjective morphism between smooth quasi-projective varieties and let $\Delta_0$ be an effective $\mathbb R$-divisor on $X_0$ such that ${{\operatorname{Supp}}}\Delta_0$ is a simple normal crossing divisor on $X_0$ and $(X_0, \Delta_0)$ is log canonical over a nonempty Zariski open set of $Y_0$. Let $L_0$ be a Cartier divisor on $X_0$ such that $L_0\sim _{\mathbb R} k(K_{X_0/Y_0}+\Delta_0)$ for some positive integer $k$. Assume that $f$ is flat. We consider the $s$-fold fiber product $$X^s_0:=\underbrace{X_0\times _{Y_0} X_0
\times _{Y_0} 