On the Effective Freeness of the Direct Images of Pluricanonical Bundles

We give effective bounds on the generation of pushforwards log-pluricanonical bundles twisted by ample line bundles. This gives a partial answer to conjecture proposed Popa and Schnell. prove two types statements: first, more in spirit general conjecture, we show generic global with predicted bound when dimesnion variety if less than 4 generally, quadratic Angehrn-Siu type bound. Secondly, assuming that relative canonical bundle is relatively semi-ample, make very precise statement. In particular, morphism smooth, it solves same bounds, for certain pluricanonical