On the uniformity of the Iitaka fibration

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration $X\dra \Ii(X)$ of smooth projective varieties of positive Kodaira dimension, provided that $\Ii(X)$ is not uniruled, the variation of the fibration is maximal, and the general fiber has a good minimal model.

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Published 1 January 2009