Generalized Nonvanishing Conjecture and Iitaka Conjecture

In this article, we will prove the Generalized Nonvanishing Conjecture holds for threefolds with either $\kappa >0$ or $q>0$. As a result, we can prove the Iitaka conjecture $C_{n,m}$ holds for $n=7$ if the source space has non-negative Kodaira dimension, if the general fibre has positive Kodaira dimension, or if the base space is not threefold with $\kappa =q=0$. In particular, $C_{n,m}^{-}$ holds if $n\le 7$.