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.
Categories: math.DG math.AP math.CV