Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an Iitaka fibration of W such that the pluricanonical system | mKW| 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.The first author was partially supported by a grant of the Leverhulme Trust. Part of this work was done when the first author visited National University of Singapore in April 2014. Part of this work was done when the first author visited National Taiwan University in August-September 2014 with the support of the Mathematics Division (Taipei Office) of the National Center for Theoretical Sciences. The visit was arranged by Jungkai A. Chen. He wishes to thank them all. The second author was partially supported by an ARF of National University of Singapore.http://dx.doi.org/10.1007/s10240-016-0080-