Strict plurisubharmonicity of Bergman kernels on generalized annuli
Volume 111 / 2014
Annales Polonici Mathematici 111 (2014), 237-243
MSC: Primary 32A25; Secondary 32U05.
DOI: 10.4064/ap111-3-2
Abstract
Let $A_\zeta=\varOmega-\overline{\rho(\zeta)\cdot\varOmega}$ be a family of generalized annuli over a domain $U$. We show that the logarithm of the Bergman kernel $K_{\zeta}(z)$ of $A_\zeta$ is plurisubharmonic provided $\rho\in {\rm PSH}(U)$. It is remarkable that $A_\zeta$ is non-pseudoconvex when the dimension of $A_\zeta$ is larger than one. For standard annuli in ${\mathbb C}$, we obtain an interesting formula for $\partial^2 \log K_{\zeta}/\partial \zeta\partial\bar{\zeta}$, as well as its boundary behavior.
