exists a positive integer $a$ such that $S^a(\tau^*\mathcal F)\otimes \tau^*\mathcal H^{\otimes -1}=
\tau^*\left(S^a(\mathcal F)\otimes 
\mathcal H^{\otimes -1}\right)$ is weakly positive (see Lemma \[x-lem4.2\]). Therefore, $S^a(\mathcal F)\otimes 
\mathcal H^{\otimes -1}$ is weakly positive since $\tau$ is finite (see, for example, [@fujino-iitaka Lemma 3.1.12 (v)]). This means that $\mathcal F$ is big by Lemma \[x-lem4.2\].

Mixed-$\omega$-sheaves and mixed-$\omega$-big sheaves {#x-sec5}
=====================================================

In this section, we introduce mixed-$\omega$-sheaves, mixed-$\omega$-big-sheaves, mixed-$\widehat\omega$-sheaves, and mixed-$\widehat\omega$-big-sheaves. We also treat some important examples in Lemmas \[x-lem5.5\], \[x-lem5.8\], and \[x-lem5.9\].

Let us start with the definition of mixed-$\omega$-sheaves and pure-$\omega$-sheaves.

\[x-def5.1\] 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 $W$.

We give a very important remark on Definition \[x-def5.1\].

\[x-rem5.2\] The notion of pure-$\omega$-sheaves is essentially the same as that of Nakayama’s $\omega$-sheaves in [@nakayama] when we treat torsion-free coherent sheaves on normal projective varieties. However, the definition of pure-$\omega$-sheaves in Definition \[x-def5.1\] does not coincide with [@nakayama Chapter 