Siu’s lemma, optimal $$L^2$$ extension and applications to twisted pluricanonical sheaves

Abstract

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 $$L^{\frac{2}{m}}$$ extension theorem for Kahler fibrations. These results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for Kahler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles.