Abstract: In birational geometry, a great deal of interest lies in the study of space of global sections (i.e. linear system) associated to a line bundle and when it produces morphisms to projective spaces (i.e. are globally generated). For instance, Takao Fujita, in 1988, conjectured that there is an effective bound on the twists of ample line bundles to obtain global generation of canonical bundles. Even though the conjecture remains unsolved as of today, partial progress was made by Angehrn-Siu, Ein-Lazarsfeld, Heier, Helmke, Kawamata, Reider, Ye-Zhu et al. In this talk I will focus on similar global generation problems for pushforwards of canonical and pluricanonical bundles under certain morphisms f: Y --> X. The canonical bundle case was first approached by Kawamata and the proof combines the Hodge theoretic properties of the morphism and the birational geometric techniques from the known cases of the Fujita conjecture. I will show how birational techniques and invariants of positivity of line bundles allow us to avoid Hodge theory to make a global generation statement on an open subset (of X) over which f behaves particularly nicely, partially proving a conjecture proposed by Popa and Schnell. This, in particular, ensures an effective non-vanishing of global sections of the pushforward. As an upshot I will discuss positivity properties of such pushforwards and an effective vanishing theorem. Finally, If time permits I will give a crash course on Kawamata's method and explain why Hodge theory is indispensable to hope for further improvements in this direction.