Vol. 313, No. 2, 2021

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

Hamid Hezari and Hang Xu

Vol. 313 (2021), No. 2, 413–432
DOI: 10.2140/pjm.2021.313.413
Abstract

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 analytic symbol 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.
Mathematical Subject Classification 2010
Primary: 32A25, 32Q15, 53C55