Let $\pi:X \longrightarrow C$ be a smooth projective morphism onto a smooth projective curve. If the general fibers are of nonnegative Kodaira dimension, is $\pi\_{\ast} \mathcal{O}(k K\_{X/C})$ nonzero for sufficiently divisible $k$? If it is, is there an algebraic proof? That is, without using invariance of plurigenera.



On direct image of the relative pluri-canonical divisor
The stalk of $\pi\_\* (\omega\_{X/C}^{\otimes k})$ at the generic point $\eta$ of $C$ equals $H^0(X\_\eta, \omega\_{X\_\eta/\kappa(\eta)}^{\otimes k})$. If the generic fiber $X\_\eta$ has nonnegative Kodaira dimension, then there exists an integer $k$ such that $H^0(X\_\eta,\omega\_{X\_\eta/\kappa(\eta)}^{\otimes mk})$ is nonzero for every positive integer $m$. Since the stalk $\pi\_\*(\omega\_{X/C}^{\otimes mk})\_\eta$ is nonzero, in particular $\pi\_\*(\omega\_{X/C}^{\otimes mk})$ is nonzero. 


