  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.
