Siu’s lemma, optimal L2 extension and applications to twisted pluricanonical sheaves

We prove a general version of Siu’s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal L 2 extension theorem and an optimal L 2 m extension theorem for Kähler fibrations. These results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for Kähler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles.