-def0.0\] A torsion-free coherent sheaf $\mathcal F$ on a normal quasi-projective variety $W$ is called a [*[mixed-$\omega$-sheaf]{}*]{} if there exist a projective surjective morphism from a smooth quasi-projective variety $V$ and a simple normal crossing divisor $D$ on $V$ such that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$. When $D=0$, $\mathcal F$ is called a [*[pure-$\omega$-sheaf]{}*]{} on $W$.  