devoted to the proof of Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\]. In Section \[x-sec11\], we treat Nakayama’s inequality on $\kappa_\sigma$, which has already played a crucial role in the theory of minimal models, and a slight generalization of the twisted weak positivity theorem.

The author was partially supported by JSPS KAKENHI Grant Numbers JP16H03925, JP16H06337. He thanks Masataka Iwai for useful comments.

We will work over $\mathbb C$, the complex number field, throughout this paper. We note that a [*[scheme]{}*]{} is a separated scheme of finite type over $\mathbb C$ and a [*[variety]{}*]{} is an integral scheme.

Preliminary {#x-sec2}
===========

In this section, we collect some basic definitions. For the details, see [@fujino-fundamental], [@fujino-foundations], and [@fujino-iitaka].

Let us start with the definition of canonical sheaves and canonical divisors.

\[x-def2.1\] Let $X$ be an equidimensional scheme of dimension $n$ and let $\omega^\bullet_X$ be the dualizing complex of $X$. Then we put $$\omega_X:=h^{-n}(\omega^\bullet_X)$$ and call it the [*[canonical sheaf]{}*]{} of $X$.

We further assume that $X$ is normal. Then a [*[canonical divisor]{}*]{} $K_X$ of $X$ is a Weil divisor on $X$ such that $$\mathcal O_{X_{\mathrm{sm}}}(K_X)\simeq 
\Omega^n_{X_{\mathrm{sm}}}$$ holds, where $X_{\mathrm{sm}}$ is the largest smooth Zariski open set of $X$.

It is well known that 