Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity

TL;DR

This paper investigates Bergman kernels associated with modules at boundary points, establishing a log-subharmonicity property that leads to concavity results and applications in effectiveness and openness conjectures.

Contribution

It introduces a new log-subharmonicity property for Bergman kernels at boundary points, providing proofs for conjectures related to effectiveness and strong openness of modules.

Findings

Proved log-subharmonicity of Bergman kernels at boundary points

Reproved sharp effectiveness result for Jonsson-Mustae1 conjecture

Established effectiveness of strong openness property for modules at boundary points

Abstract

In this article, we consider Bergman kernels with respect to modules at boundary points, and obtain a log-subharmonicity property of the Bergman kernels, which deduces a concavity property related to the Bergman kernels. As applications, we reprove the sharp effectiveness result related to a conjecture posed by Jonsson-Must\c{a}t\u{a} and the effectiveness result of strong openness property of the modules at boundary points.