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

TL;DR

This paper provides an elementary proof of the asymptotic expansion of the Bergman kernel near the diagonal by showing it as a perturbation of the Bargmann-Fock kernel after rescaling the Kähler potential.

Contribution

It introduces a new elementary method to prove the asymptotic expansion of the Bergman kernel using perturbation of the Bargmann-Fock model.

Findings

Bergman kernel asymptotics can be derived via elementary perturbation methods.

Rescaling the Kähler potential reveals the kernel as a perturbation of the Bargmann-Fock kernel.

The approach simplifies existing proofs of the asymptotic expansion.

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.