  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.
