Asymptotic Expansion of the Bergman Kernel via Perturbation of the Bargmann–Fock Model

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated with the kth tensor powers of a positive line bundle L in a 1k-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential kφ in a 1k-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.