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 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. 
Hamid Hezari, Casey Lynn Kelleher, Shoo Seto, Hang Xu
math.DG math.AP math.CV