Parabolicity of the regular locus of complex varieties

TL;DR

This paper proves that the regular locus of complex varieties is locally parabolic at singular points, leading to implications for the $L^2$-theory of the $ar{d}$-operator on singular complex spaces.

Contribution

It establishes the local parabolicity of the regular locus at singularities and applies this to $L^2$-theory for the $ar{d}$-operator on singular spaces.

Findings

Regular locus is locally parabolic at singular points

Regular locus of compact complex varieties is parabolic

Application to $L^2$-theory for the $ar{d}$-operator

Abstract

The purpose of this note is to show that the regular locus of a complex variety is locally parabolic at the singular set. This yields that the regular locus of a compact complex variety, e.g., of a projective variety, is parabolic. We give also an application to the $L^2$-theory for the $\overline{\partial}$-operator on singular spaces.