Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness

TL;DR

This paper provides new quantitative characterizations of Griffiths positivity, pluriharmonicity, and flatness of Hermitian vector bundles and metrics using $L^2$-extension conditions, confirming a conjecture and extending previous results.

Contribution

It introduces a quantitative framework for Griffiths positivity and establishes equivalences for pluriharmonicity and flatness via $L^p$-extension conditions, generalizing prior work.

Findings

Griffiths positivity characterized by $L^2$-extension conditions

Pluriharmonic functions characterized by equality in $L^p$-extension condition

Flatness of Hermitian metrics characterized by $L^p$-extension equality

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.