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On a property of
Bergman kernels when the Kähler potential is analytic
Hamid Hezari and Hang Xu |
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Vol. 313 (2021), No. 2, 413–432
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DOI: 10.2140/pjm.2021.313.413
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Abstract |
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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
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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 (PDF Access Denied
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Keywords
Kahler manifold, positive line bundle, the Bergman kernel.
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Mathematical Subject Classification 2010
Primary: 32A25, 32Q15, 53C55
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Milestones
Received: 24 December 2019
Revised: 22 March 2021
Accepted: 31 May 2021
Published: 12 October 2021
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Authors |
| Hamid Hezari | |
| Department of Mathematics University of California, Irvine Irvine, CA United States |
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| Hang Xu | |
| Department of Mathematics University of California, San Diego La Jolla, CA United States |
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