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We prove the decomposition theorem for Hodge modules with integral structurealong proper K\"ahler morphisms, partially generalizing M. Saito's theorem forprojective morphisms. Our proof relies on compactifications of period maps ofgenerically defined variations of Hodge structure, as well as the theorems ofCattani-Kaplan-Schmid on \(L^2\)-cohomology of variations of Hodge structurefor the hard Lefschetz theorem.

Hodge modules and K\"ahler morphisms