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