On a property of Bergman kernels when the Kähler potential is analytic

Abstract

We provide a simple proof of a result of Rouby-Sjöstrand-Ngoc \cite{RSN} and Deleporte \cite{Deleporte}, which asserts that if the Kähler 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 Kähler 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}.

arXiv admin note: text overlap with arXiv:1705.09281

References in corpus (3)

• Analytic Bergman operators in the semiclassical limit
• Quantitative upper bounds for Bergman kernels associated to smooth Kähler potentials
• The Bergman kernel in constant curvature

Cited by in corpus (2)

• A direct approach to the analytic Bergman projection
• WKB eigenmode construction for analytic Toeplitz operators