Nakano positivity of singular Hermitian metrics: Approximations and applications

This paper studies the approximation of singular Hermitian metrics on vector bundles using smooth Hermitian metrics with Nakano semi-positive curvature on Zariski open sets. We show that singular Hermitian metrics capable of this approximation satisfy Nakano semi-positivity as defined through the ∂̄-equation with optimal L²-estimates. Furthermore, for a projective fibration f X → Y with a line bundle L on X, we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf f*OX(KX/Y+L) admits this approximation. As an application, we establish several vanishing theorems.