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.