W$ and $H'$ such that $\tau^*H\sim dH'$. By the construction of the above Kawamata cover $\tau:W'\to W$, we may assume that the ramification locus $\Sigma$ of $\tau$ in $W$ is a general simple normal crossing divisor. This means that $f^*P$ is a smooth divisor for any irreducible component $P$ of $\Sigma$ and that $f^*\Sigma$ is a simple normal crossing divisor on $V$. In this situation, we can easily check that $V'=V\times _WW'$ is a smooth quasi-projective variety. $$\xymatrix{
 V' \ar[r]^{\rho} \ar[d]_{f'} & V\ar[d]^{f} \\
  W' \ar[r]_{\tau} & W
}$$ By construction, we can also easily check that $\omega_{V'/W'}=\rho^*\omega_{V/W}$ by the Hurwitz formula.

Let us see the construction of $f':V'\to W'$ more precisely for the reader’s convenience. Let $\mathcal A$ be an ample invertible sheaf on $W$ such that $\mathcal A^{\otimes d}\otimes \mathcal O_W(-D_i)$ is generated by global sections for $i=1, 2$. We put $n=\dim W$. We take smooth divisors $$H^{(1)}_1, \ldots, H^{(1)}_n, H^{(2)}_1, 
\ldots, H^{(2)}_n$$ on $W$ in general position such that $\mathcal A^{\otimes d}= \mathcal O_W(D_i+H^{(i)}_j)$ for $1\leq j\leq n$ and $i=1, 2$. Let $Z^{(i)}_j$ be the cyclic cover associated to $\mathcal A^{\otimes d}= \mathcal O_W(D_i+H^{(i)}_j)$ for $1\leq j\leq n$ and $i=1, 2$. Then $W'$ is the normalization of 