Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf F endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal ex- tension property admits a direct-sum decomposition F ≃ U \oplus A where U is a hermitian flat bundle and A is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles f_*(\omega_{X/Y}^{\otimes m}) under a surjective morphism f : X → Y of smooth projective varieties with m ≥ 2. This extends previous results of Fujita, Catanese–Kawamata, and Iwai.