Multiplier Submodule Sheaves and a problem of Lempert

TL;DR

This paper proves an $L^2$ extension theorem for Nakano semi-positive singular Hermitian metrics, demonstrating stability of multiplier submodule sheaves and affirmatively resolving Lempert's question on Nakano semi-positivity preservation under limits.

Contribution

It introduces new stability results for multiplier submodule sheaves and solves a longstanding question about Nakano semi-positivity limits.

Findings

Established an $L^2$ extension theorem for Nakano semi-positive singular Hermitian metrics.

Proved strong openness and stability of multiplier submodule sheaves.

Confirmed preservation of Nakano semi-positivity under increasing limits.

Abstract

In this article, we establish an $L^2$ extension theorem for Nakano semi-positive singular Hermitian metrics on holomorphic vector bundles, and the strong openness and stability properties of the multiplier submodule sheaves associated to Nakano semi-positive singular Hermitian metrics on holomorphic vector bundles. We solve affirmatively a question of Lempert on the preservation of Nakano semi-positivity under limit of an increasing metrics based on Deng-Ning-Wang-Zhou's characterization of Nakano positivity.