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

Abstract

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).