Fiberwise Bergman kernels, vector bundles, and log-subharmonicity

TL;DR

This paper investigates the log-subharmonicity of Bergman kernels associated with vector bundles and modules at boundary points, leading to new estimates and reproofs of key properties in complex analysis.

Contribution

It introduces a log-subharmonicity property for Bergman kernels related to modules at boundary points for singular hermitian metrics on vector bundles, providing new estimates and proofs.

Findings

Established log-subharmonicity of Bergman kernels for singular hermitian metrics.

Derived lower bounds for weighted L^2 integrals on sublevel sets.

Reproved the strong openness property of modules.

Abstract

In this article, we consider Bergman kernels related to modules at boundary points for singular hermitian metrics on holomorphic vector bundles, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted $L^2$ integrals on sublevel sets of plurisubharmonic functions, and reprove an effectiveness result of the strong openness property of the modules.