Multiplier submodule sheaves, 𝐿 ² extension and a problem of Lempert

In this article, we establish an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L² extension theorem for Nakano semi-positive singular Hermitian metrics on holomorphic vector bundles, and the strong openness and stability properties for the multiplier submodule sheaves associated to such singular metrics. We solve affirmatively a question of Lempert on the preservation of Nakano semi-positivity under limit of an increasing sequence of metrics based on Deng–Ning–Wang–Zhou’s characterization of Nakano positivity.