On a property of Bergman kernels when the K\"ahler potential is analytic

TL;DR

This paper proves that when the K"ahler potential is real analytic, the Bergman kernel exhibits an analytic structure with an asymptotic expansion and exponentially small remainder, simplifying previous proofs.

Contribution

It provides a simpler proof that the Bergman kernel is analytic when the K"ahler potential is real analytic, using a recursive formula for the kernel coefficients.

Findings

Bergman kernel is an analytic kernel under real analytic K"ahler potential.

The kernel admits an asymptotic expansion near the diagonal with exponentially small remainder.

The proof simplifies previous approaches by using a recursive formula for coefficients.

Abstract

We provide a simple proof of a result of Rouby-Sj\"ostrand-Ngoc \cite{RSN} and Deleporte \cite{Deleporte}, which asserts that if the K\"ahler potential is real analytic then the Bergman kernel is an \textit{analytic kernel} meaning that its amplitude is an \textit{analytic symbol} and its phase is given by the polarization of the K\"ahler potential. This in particular shows that in the analytic case the Bergman kernel accepts an asymptotic expansion in a fixed neighborhood of the diagonal with an exponentially small remainder. The proof we provide is based on a linear recursive formula of L. Charles \cite{Cha03} on the Bergman kernel coefficients which is similar to, but simpler than, the ones found in \cite{BBS}.