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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 ...

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension). Let $g:X\rightarrow ...