A variational formula for Bergman kernels

For a given family of domains $\Omega_t\subset\mathbb C^n$, $t\in[0,1]$, under some assumptions a formula for $B_1(z,s)-B_0(z,s)$ is established, where $B_0$ and $B_1$ are the Bergman kernels for $\Omega_0$ and $\Omega_1$. As an application of this formula, we obtain two terms in the asymptotics of $B(z,z)$ as $z\to\partial\Omega$ for a special class of domains.