Continuous L2-optimal Hermitian metrics satisfy an optimal L2 extension inequality with the logarithmic capacity constant, which implies they are Griffiths semi-positive.
Multiplier Submodule Sheaves and a problem of Lempert
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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.
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An optimal $L^2$ extension for continuous $L^2$-optimal Hermitian metrics
Continuous L2-optimal Hermitian metrics satisfy an optimal L2 extension inequality with the logarithmic capacity constant, which implies they are Griffiths semi-positive.