For every smooth complex projective variety w of  dimension and nonnegative Kodaira dimension, we show the existence of a universal constant  depending only on and two natural invariants of the general fibres of an Iitaka fibration of such that the pluricanonical system |mK_W|defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.

This is a joint work with C. Birkar.

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