The ∂̄-cohomology groups, holomorphic morse inequalities, and finite type conditions

We study spectral behavior of the complex Laplacian on forms with values in the k^th tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. We establish upper estimates of the dimensions of the Dolbeault cohomology groups on forms with values in the power of the line bundle. We prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant C, the number of eigenvalues of the ∂̄-Neumann Laplacian less than or equal to Ck has at most polynomial growth as k tends to infinity.