Iitaka fibration over canonical model

Question

I am looking for a referrence for the proof of following fact

If the minimal projective manifold has positive Kodaira dimension and it is not of general type, it admits an Iitaka fibration over its canonical model

Answer 1

In Lazarsfeld, Positivity in Algebraic Geometry I, Section 2.1, one can find a proof the following:

Theorem 2.1.33: For any line bundle $L$ on a normal variety $X$ such that $\kappa(X,L)>0$ there is an associated rational map $\phi_L : X \dashrightarrow Y$ such that $\operatorname{dim} Y=\kappa(X,L)$ and (after passing to a resolution of $\phi$) the restriction of $L$ to a very general fibre of $\phi_L$ has Iitaka dimension zero. $\phi_L$ is called the Iitaka fibration associated to $L$.

Although Lazarsfeld doesn't say so explicity, in fact this $Y$ is Proj of the section ring $R(X,L)$ of $L$.

Now apply this with $L=K_X$. Since the canonical model of $X$ is by definition $\operatorname{Proj}R(X,K_X)$, the theorem above gives exactly what you want.

The hypothesis "not of general type" seems to be superfluous, although it does ensure that the Iitaka fibration is nontrivial. So if you require "fibration" to include the condition "positive-dimensional fibres", then you should keep this.