Applications of the Ohsawa-Takegoshi Extension Theorem to Direct Image Problems

In the 1st part of the paper, we study a Fujita-type conjecture by Popa and Schnell and obtain an effective bound on the generic global generation of direct images of twisted pluricanonical bundles. We also point out the relation between the Seshadri constant and the optimal bound. In the 2nd part, we give an affirmative answer to a question by Demailly, Peternell, and Schneider in a more general setting. As byproducts of our proofs, we extend a result by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal settings and refine a theorem by Broustet and Pacienza on the rational connectedness of the image.