Spectrum of the complex Laplacian on product domains

TL;DR

This paper demonstrates that the spectrum of the complex Laplacian on a product of hermitian manifolds is the Minkowski sum of the spectra on individual factors, and uses this to analyze the d-bar operator's range.

Contribution

It establishes a spectral decomposition for the complex Laplacian on product manifolds and applies this to the closedness of the d-bar operator's range.

Findings

Spectrum of complex Laplacian on product manifolds is Minkowski sum of factors' spectra.

Range of the d-bar operator on a product is closed if it is closed on each factor.

Provides a spectral analysis tool for complex geometry on product domains.

Abstract

We show that the spectrum of the complex Laplacian on a product of hermitian manifolds is the Minkowski sum of the spectra of the complex Laplacians on the factors. We use this fact to show that the range of the d-bar operator on a product manifold is closed, provided it is closed in each factor.