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

For high power $k$, the $L^2$-estimates for the Dirac-Dolbeault operator with coefficient $L^k\otimes E$ can be obtained from the Bochner-Kodaira-Nakano identity if $L$ has positive curvature. In this article, we generalize the classical method to obtain $L^2$-estimates for mixed curvature case, and give a bound to the extra error term. Modifying the $L^2$-estimates and existence theorems for $\bar{\mathrm{\partial }}$-operator, we can get a local spectral gap of the Kodaira Laplacian $\mathrm{\square }$ and thus a full asymptotic expansion for Bergman kernel.