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 $\bar{\mathrm{\partial }}$-equation with optimal $L^2$-estimates. Furthermore, for a projective fibration $f\text{\hspace{0.17em}⁣}:X\to Y$ with a line bundle $L$ on $X$, we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf $f_{\ast }{\mathcal{O}}_X(K_{X\mathrm{/}Y}+L)$ admits this approximation. As an application, we establish several vanishing theorems.