Optimal $L^2$ extension for holomorphic vector bundles with singular hermitian metrics

TL;DR

This paper establishes an optimal $L^2$ extension theorem for holomorphic vector bundles with singular hermitian metrics on weakly pseudoconvex Kähler manifolds, advancing the understanding of singular positivity and extension conditions.

Contribution

It introduces a new optimal $L^2$ extension theorem for singular hermitian vector bundles and explores conditions for equality and applications to existing extension theorems.

Findings

Characterization of singular Nakano positivity

Necessary conditions for equality in extension theorem

Extensions of $L^2$ theorems to singular hermitian bundles

Abstract

In the present paper, we study the properties of singular Nakano positivity of singular hermitian metrics on holomorphic vector bundles, and establish an optimal $L^2$ extension theorem for holomorphic vector bundles with singular hermitian metrics on weakly pseudoconvex K\"{a}hler manifolds. As applications, we give a necessary condition for the holding of the equality in optimal $L^2$ extension theorem, and present singular hermitian holomorphic vector bundle versions of some $L^2$ extension theorems with optimal estimate.