Nakano positivity of singular Hermitian metrics: Approximations and applications

Abstract

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 \(\overline{\partial} \)-equation with optimal \(L^2\)-estimates. Furthermore, for a projective fibration \(f \colon X \to Y\) with a line bundle \(L\) on \(X\), we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf \(f_{*}\mathcal{O}_{X}(K_{X/Y}+L)\) admits this approximation. As an application, we establish several vanishing theorems.