Hodge modules and K\"ahler morphisms

We prove the decomposition theorem for Hodge modules with integral structure along proper K\"ahler morphisms, partially generalizing M. Saito's theorem for projective morphisms. Our proof relies on compactifications of period maps of generically defined variations of Hodge structure, as well as the theorems of Cattani-Kaplan-Schmid on $L^2$-cohomology of variations of Hodge structure for the hard Lefschetz theorem.