A canonical bundle formula for certain algebraic fiber spaces and its applications

Abstract

We investigate period maps of polarized variations of Hodge structures of weight one or two. We treat the case when the period domains are bounded symmetric domains. We deal with a relationship between canonical extensions of some Hodge bundles and automorphic forms. As applications, we obtain a canonical bundle formula for certain algebraic fiber spaces, such as Abelian fibrations, $K3$ fibrations, and solve Iitaka's famous conjecture $C_{n, m}$ for some algebraic fiber spaces.