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

We prove that a torsion-free sheaf F \mathcal {F} endowed with singular hermitian metric semi-positive curvature and satisfying the minimal extension property admits direct-sum decomposition F asymptotically-equals script U circled-plus A"> ≃ mathvariant="script">U ⊕ X / Y ⊗<!-- ⊗ <mml:mi>m encoding="application/x-tex">f_* \omega _{X/Y}^{\otimes m} under surjective morphism colon right-arrow Y"> :<!-- : <mml:mo stretchy="false">→<!-- → encoding="application/x-tex">f\colon \to Y smooth projective varieties alttext="m greater-than-or-equal-to 2"> ≥<!-- ≥ <mml:mn>2 encoding="application/x-tex">m\geq 2 . This extends previous results Fujita, Catanese–Kawamata, Iwai.