
> 
> Let $\pi: X\to Y$ be an Iitaka fibration of projective varieties
>  $X,Y$, then is there always the following decomposition
> 
> 
> $$K\_Y+\frac{1}{m!}\pi\_\*\mathcal O\_X(m!K\_{X/Y})=P+N$$ 
> 
> 
> where $P$ is semiample and $N$ is effective and 
> 
> 
> $$H^0(Y,\mathcal O\_X(maP))\cong H^0(X,\mathcal O(ma(K\_X+\frac{1}{m!}\pi\_\*\mathcal O\_X(m!K\_{X/Y}))))$$
> 
> 
> holds for every $m\geq 0$ and some positive integer $a$ and $P,N\in
>  Div(X)$?
> 
> 
> I think, such decomposition is correct due to Higher canonical bundle
>  formula of Fujino-Mori, but I am not sure.
> 
> 
> 



A Decomposition for Iitaka fibration
If $X$ is smooth (projective over the complex numbers), then $R(K\_X)$ is finitely generated by BCHM. We may thus assume that $R(kK\_X)$ is generated in degree 1 for some $k>0$. Passing to a log resolution of $|kK\_X|$ we may assume that $|kK\_X|=M+F$ where $F$ is the fixed divisor and $M$ is base point free and so $M$ defines a morphism $f:X\to Y$ which is the Iitaka fibration. Thus $M=f^\*O\_Y(1)$ is semiample and $F$ is effective, and $H^0(O\_Y(m))=H^0(O\_X(mM))=H^0(O\_X(mkK\_X))$.
Fujino-Mori's canonical bundle formula (their theorem 5.2) is in fact used in the proof of the finite generation of $R(K\_X)$ by relating $R(K\_X)$ to $R(K\_Y+B\_Y)$ where $(Y,B\_Y)$ is klt, $K\_Y+B\_Y$ is big and $X\to Y$ is (a birational model of) the Iitaka fibration. 


