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

TL;DR

This paper links the finite type condition of pseudoconvex domains in complex analysis to the spectral growth of the $ar{ ext{d}}$-Neumann Laplacian on tensor powers of line bundles, providing a spectral characterization in two dimensions.

Contribution

It establishes a new equivalence between finite type boundary conditions and polynomial eigenvalue growth of the complex Laplacian in two-dimensional cases.

Findings

Finite type domains correspond to polynomial eigenvalue growth.

Spectral behavior of the Laplacian characterizes boundary regularity.

Results are specific to two-dimensional complex manifolds.

Abstract

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.