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

Abstract

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