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

Semi-Classical Asymptotics of Bergman and Spectral Kernels for (0,q)-forms