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

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-Mustatȃ and the effectiveness result of strong openness property of the modules at boundary points.

Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity II -- on Stein manifolds

“…Corollary 1.6 (see [31,1]). Let ϕ be a plurisubharmonic function on M , and let f be a holomorphic (n, 0) form on M t0 = {Ψ < −t 0 } for some t 0 ≥ T such that f ∈ A 2 (M t0 , e −ϕ ).…”

“…In [1], we considered Bergman kernels related to modules at boundary points on pseudoconvex domains, and obtained a log-subharmonicity property of the Bergman kernels, which deduces a new approach from Conjecture J-M to the strong openness property.…”

“…When M is a pseudoconvex domain in C n , Z 0 is one point, and T = 0, Theorem 1.3 can be referred to [1]. If F (z) = 0 for some z ∈ M , set Ψ(z) = −T .…”

Fiberwise Bergman kernels, vector bundles, and log-subharmonicity

“…In [1] (see also [2]), we considered Bergman kernels related to modules at boundary points of the sub-level sets, and obtained the log-subharmonicity property of the Bergman kernels. We applied the log-subharmonicity to get a lower estimate of weighted L 2 integrals on sublevel sets, and reproved the effectiveness result of strong openness property of modules at boundary points.…”