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

Fiberwise Bergman kernels, vector bundles, and log-subharmonicity