The $\overline\partial$-cohomology groups, holomorphic Morse inequalities, and finite type conditions

Details The $\overline\partial$-cohomology groups, holomorphic Morse inequalities, and finite type conditions The $\overline\partial$-cohomology groups, holomorphic Morse inequalities, and finite type conditions

2007-12-07

arxiv.org/abs/0712.1218v1

Mathematics

Complex Variables

27 pages

Scientific paper

We 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.

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