On the singularities of the spectral and Bergman projections on complex manifolds with boundary

We show that the spectral kernel of the N @-Neumann Laplacian acting on .0;q/-forms on a smooth relatively compact domain admits a full asymptotic expansion near the nondegenerate part of the boundary.We show further that the Bergman projection admits an asymptotic expansion under certain local closed range condition.In particular, if condition Z.q/ fails but conditions Z.q 1/ and Z.q C 1/ hold, the Bergman projection on .0;q/-forms admits an asymptotic expansion.As applications, we establish Bergman kernel asymptotic expansions near nondegenerate points of some domains with weakly pseudoconvex boundary and S 1 -equivariant asymptotic expansions and embedding theorems for domains with holomorphic S 1 -action.1. Introduction 409 2. Preliminaries 421 3. The boundary operator .q/427 4. Parametrices for the N @-Neumann Laplacian outside the critical degree 432 5. Microlocal Hodge decomposition in the critical degree 439 6. Microlocal spectral theory for the N @-Neumann Laplacian 460 7. Proof of Theorem 1.9 468 8. S 1 -equivariant Bergman kernel asymptotics and embedding theorems 469