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 $W$.

For the study of klt pairs, the notion of pure-$\omega$-sheaves is sufficient and is essentially due to Nakayama (see [@nakayama]). In this paper, we study some basic properties of mixed-$\omega$-sheaves. They are indispensable for the study of log canonical pairs. Of course, the theory of mixed-$\omega$-sheaves (resp. pure-$\omega$-sheaves) in this paper is based on that of mixed (resp. pure) Hodge structures. Roughly speaking, Nakayama only treats pure-$\omega$-sheaves in [@nakayama Chapter V]. However, his theory of $\omega$-sheaves is more sophisticated and some of his results are much sharper than ours. We do not try to make the framework discussed in this paper supersede Nakayama’s theory of $\omega$-sheaves in [@nakayama Chapter V]. The main purpose of this paper is to make Nakayama’s theory of $\omega$-sheaves more accessible and make it applicable to the study of log canonical pairs. Theorem \[x-thm9.3\] (and Remark \[x-rem9.4\]) is one of the main results of this paper, which we call a 