  Let $M$ be a complex manifold of dimension $n$ with smooth boundary $X$.
Given $q\in\{0,1,\ldots,n-1\}$, let $\Box^{(q)}$ be the $\ddbar$-Neumann
Laplacian for $(0,q)$ forms. We show that the spectral kernel of $\Box^{(q)}$
admits a full asymptotic expansion near the non-degenerate part of the boundary
$X$ and the Bergman projection admits an asymptotic expansion under some local
closed range condition. As applications, we establish Bergman kernel asymptotic
expansions for some domains with weakly pseudoconvex boundary and
$S^1$-equivariant Bergman kernel asymptotic expansions and embedding theorems
for domains with holomorphic $S^1$-action.
