Geodesic
A variational formula for Bergman kernels
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 221-243 
N. A. Shirokov. A variational formula for Bergman kernels. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 221-243. http://geodesic.mathdoc.fr/item/ZNSL_1998_255_a14/
@article{ZNSL_1998_255_a14,
     author = {N. A. Shirokov},
     title = {A variational formula for {Bergman} kernels},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {221--243},
     year = {1998},
     volume = {255},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_255_a14/}
}
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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.