Plurisubharmonicity of Bergman Kernels on generalized annuli

Abstract

Let $A_{\zeta }=\Omega -\overline{\rho (\zeta )\cdot \Omega }$ be a family of generalized annuli over a domain $U$. We show that the logarithm $\log K_{\zeta }(z)$ of the Bergman kernel $K_{\zeta }(z)$ of $A_{\zeta }$ is plurisubharmonic provided $\rho \in PSH(U)$. 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.