Bergman kernels for almost spherical domians with $H^{\sigma}$-smoth boundaries

This paper is devoted to the behavior of the Bergman kernels for almost spherical striclty pseudoconvex domians in $\mathbb C^n$. We find first several terms of the asymptotics for the Bergman kernel of striclty pseudoconvex domian with $H^{\sigma}$-smooth boundary, $4<\sigma<6$. New terms in comparison with the case of $C^6$-smoth boundary appear but the growth rate of the remainder remains the same.