  In this paper, we develop a new scaling method to study spectral and Bergman
kernels for the k-th tensor power of a line bundle over a complex manifold
under local spectral gap condition. In particular, we establish a simple proof
of the pointwise asymptotics of spectral and Bergman kernels. As a new result,
in the function case, we obtain the leading term of Bergman kernel under
spectral gap with exponential decay. Moreover, in the general cases of
(0,q)-forms, the asymptotics remain valid while the curvature of the line
bundle is degenerate.
