Asymptotic expansion of the Bergman kernel via perturbation of the Bargmann-Fock model

Hamid Hezari; Casey Lynn Kelleher; Shoo Seto; Hang Xu

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arXiv · 1411.7438

Asymptotic expansion of the Bergman kernel via perturbation of the Bargmann-Fock model

Hamid Hezari·Casey Lynn Kelleher·Shoo Seto·Hang Xu

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Abstract

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the K\"ahler potential $k\varphi$ in a $\frac{1}{\sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.

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BibTeX RIS

Hamid Hezari, Casey Lynn Kelleher, Shoo Seto, Hang Xu. 2014-11-27. Asymptotic expansion of the Bergman kernel via perturbation of the Bargmann-Fock model. https://arxiv.org/abs/1411.7438

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