Optimal \(L^2\) Extension for Holomorphic Vector Bundles with Singular Hermitian Metrics

In the present paper, we study the properties of singular Nakano positivity of singular Hermitian metrics on holomorphic vector bundles, and establish an optimal \(L^2\) $L^2$ extension theorem for holomorphic vector bundles with singular Hermitian metrics on weakly pseudoconvex Kähler manifolds, which is a unified version of the optimal \(L^2\) $L^2$ extension theorems for holomorphic line bundles with singular Hermitian metrics of Guan–Zhou and Zhou–Zhu. As applications, we give a necessary condition for the holding of the equality in optimal \(L^2\) $L^2$ extension theorem, and present singular Hermitian holomorphic vector bundle versions of some \(L^2\) $L^2$ extension theorems with optimal estimate.