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For high power $k$, the $L^2$-estimates for the Dirac-Dolbeault operator withcoefficient $L^k\otimes E$ can be obtained from the Bochner-Kodaira-Nakanoidentity if $L$ has positive curvature. In this article, we generalize theclassical method to obtain $L^2$-estimates for mixed curvature case, and give abound to the extra error term. Modifying the $L^2$-estimates and existencetheorems for $\bar{\partial}$-operator, we can get a local spectral gap of theKodaira Laplacian $\Box$ and thus a full asymptotic expansion for Bergmankernel.

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds