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

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

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

纤维方向的Bergman核、向量丛与对数次调和性

Fiberwise Bergman kernels, vector bundles, and log-subharmonicity

本文研究全纯向量丛上奇异Hermitian度量边界点处模相关的Bergman核，证明其对数次调和性，并应用于加权$L^2$积分下界估计及强开性有效性结果的重新证明。

Curvature of vector bundles and subharmonicity of Bergman kernels

In a previous paper, \cite{Berndtsson}, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted $L^2$-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of $L^2$-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle, $V$, over a complex manifold is ample in the sense of Hartshorne, then $V\gr…

Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $ϕ$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z; (t,z)\in D\}$, let $ϕ^t$ be the restriction of $ϕ$ to $D_t$ and denote by $K_t(z,ζ)$ the Bergman kernel of $D_t$ with the weight function $ϕ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $ϕ=0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setti…

变分泛函的纤维式ξ-Bergman核的对数多重次调和性

The log-plurisubharmonicity of fiberwise $ξ-$Bergman kernels for variant functionals

本研究建立了变分泛函的纤维式ξ-Bergman核的对数多重次调和性，解答了相关公开问题，并应用该结论证明多圆盘上满足特定乘子理想层包含关系的纤维点集为多重极集。

退化情形下的 Bergman 核

Bergman kernels on degenerations

本文研究极化簇平坦退化中的 Bergman 核，定义逐纤维 Bergman 核并证明其连续性及 Fubini–Study 流的一致收敛性。