On a property of Bergman kernels when the K\"ahler potential is analytic
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}.
Cite
@article{arxiv.1912.11478,
title = {On a property of Bergman kernels when the K\"ahler potential is analytic},
author = {Hamid Hezari and Hang Xu},
journal= {arXiv preprint arXiv:1912.11478},
year = {2021}
}
Comments
arXiv admin note: text overlap with arXiv:1705.09281