fundamental theorem of the theory of mixed-$\omega$-sheaves.

\[y-thm1.10\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $D$ be an $\mathbb R$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 2$. Assume the following conditions:

-   $(X, \Delta)$ is log canonical [*[(]{}*]{}resp. klt[*[)]{}*]{} over a nonempty Zariski open set of $Y$, and

-   $L+f^*D-k(K_{X/Y}+\Delta)-f^*A$ is semi-ample for some big $\mathbb R$-divisor $A$ on $Y$.

If $f_*\mathcal O_Y(L)\ne 0$, then there exist a mixed-$\omega$-big-sheaf [*[(]{}*]{}resp. pure-$\omega$-big-sheaf[*[)]{}*]{} $\mathcal F$ on $Y$ and a generically isomorphic injection $$\mathcal F\hookrightarrow 
\mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L).$$

For the precise definition of mixed-$\omega$-big-sheaves and pure-$\omega$-big-sheaves, see Definition \[x-def5.3\] below.

As an application of Theorem \[y-thm1.10\], we have:

\[y-thm1.11\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with connected fibers. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of 