Parabolicity of the regular locus of complex varieties

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.