Siu’s lemma: Generalizations and applications

Abstract

In this survey paper, we present some generalizations of Siu’s lemma related to multiplier ideal sheaves and discuss their applications in some problems related to optimal $L^2$ extension, comparison between singular metrics on exceptional fibers of twisted relative pluricanonical bundles, and subadditivity of Kodaira–Iitaka dimensions with multiplier ideal sheaves. We also discuss some ideas in the proofs.