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We present a simplified proof for a recent theorem by Junyan Cao and MihaiPaun, confirming a special case of Iitaka's conjecture: if $f \colon X\to Y$ isan algebraic fiber space, and if the Albanese mapping of $Y$ is genericallyfinite 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 analytictechniques, especially the construction of singular hermitian metrics onpushforwards of adjoint bundles (due to Berndtsson, Paun, and Takayama).

Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun