Subadditivity of generalized Kodaira dimensions and extension theorems

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an [Formula: see text] extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal [Formula: see text] extension theorems with singular weights on weakly pseudoconvex Kähler manifolds.