2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143796We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, 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 $\overline\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity.27 pagesComplex VariablesDifferential Geometry32L10; 32W05text