Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness

Zhuo Liu; Wang Xu

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arXiv · 2210.17361

Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness

Zhuo Liu·Wang Xu

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Abstract

Deng-Ning-Wang-Zhou showed that a Hermitian holomorphic vector bundle is Griffiths semi-positive if it satisfies the optimal $L^2$-extension condition. As a generalization, we present a quantitative characterization of Griffiths positivity in terms of certain $L^2$-extension conditions. We also show that a $\mathbb{R}$-valued measurable function is pluriharmonic if and only if it satisfies the equality part of the optimal $L^p$-extension condition. This answers a conjecture of Inayama affirmatively. Moreover, the flatness of a possibly singular Hermitian metric is also equivalent to the equality part of the optimal $L^p$-extension condition.

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Zhuo Liu, Wang Xu. 2022-10-31. Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness. https://arxiv.org/abs/2210.17361

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