  We present a simplified proof for a recent theorem by Junyan Cao and Mihai
Paun, confirming a special case of Iitaka's conjecture: if $f \colon X\to Y$ is
an algebraic fiber space, and if the Albanese mapping of $Y$ is generically
finite over its image, then we have the inequality of Kodaira dimensions
$\kappa (X)\geq \kappa (Y)+\kappa (F)$, where $F$ denotes a general fiber of
$f$. We include a detailed survey of the main algebraic and analytic
techniques, especially the construction of singular hermitian metrics on
pushforwards of adjoint bundles (due to Berndtsson, Paun, and Takayama).
