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.
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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. 

