seminar

The $\overline{\partial }$ -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 smooth bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in 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 d-bar-Neumann Laplacian less than or equal to Ck has at most polynomial growth as k goes to infinity. This talk is based on a joint paper with Howard Jacobowitz.