We provide a simple proof of a result of Rouby–Sjöstrand–Ngoc (2020) and
Deleporte (2021), which asserts that if the Kähler potential is real analytic then the
Bergman kernel is an
analytic kernel, meaning that its amplitude is an
analyticsymbol and its phase is given by the polarization of the
otential. 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
(2003) on the Bergman kernel coefficients which is similar to, but simpler than, the
ones found by Berman, Berndtsson, and Sjöstrand in (Ark. Mat. 46:2 (2008),
197–217).
Keywords
Kahler manifold, positive line bundle, the Bergman kernel.
otential. 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
(2003) on the Bergman kernel coefficients which is similar to, but simpler than, the
ones found by Berman, Berndtsson, and Sjöstrand in (