Compactness of the d-bar-Neumann operator on singular complex spaces

TL;DR

This paper investigates the conditions under which the d-bar-Neumann operator is compact on singular complex spaces, providing new criteria and construction methods for compact solution operators in the presence of isolated singularities.

Contribution

It introduces a novel criterion for the compactness of the d-bar-Neumann operator on singular spaces and constructs compact solution operators based on this criterion.

Findings

Compactness of the d-bar-Neumann operator at isolated singularities for certain degrees.

Construction of compact solution operators for the d-bar-equation on singular spaces.

A general characterization of compactness in function spaces on singular complex spaces.

Abstract

Let X be a Hermitian complex space of pure dimension n. We show that the d-bar-Neumann operator on (p,q)-forms is compact at isolated singularities of X if q>0 and p+q is not equal to n-1 or n. The main step is the construction of compact solution operators for the d-bar-equation on such spaces which is based on a general characterization of compactness in function spaces on singular spaces, and that leads also to a criterion for compactness of more general Green operators on singular spaces.