We generalize Radó’s extension theorem from the complex plane to reduced complex spaces.

Let X be a complex n-dimensional reduced analytic space with isolated singular point x0, and with a strongly plurisubharmonic function ρ : X → [0, ∞) such that ρ(x0) = 0. A smooth Kähler form on X \ {x0} is then defined by i∂∂ρ. The associated metric is assumed to have -curvature, to admit the Sobolev inequality and to have suitable volume growth near x0. Let E → X \ {x0} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0, 1)-form with coefficients in E. The main result of this article states that if ξ and the curvature of E are both then the equation has a smooth solution on a punctured neighbourhood of x0. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of ${{\mathbb{C}}^{N}}\left( N\ge 2 \right)$ is removable for holomorphic functions, if its topological dimension is less than or equal to $N\,-\,2$. Besides, they asked whether closed subsets of ${{\mathbb{C}}^{2}}$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question.

We continue our research on extension of complex varieties across closed subsets. While efforts are being made to deal with varieties of any dimensions, the paper primarily concerns 1-dimensional case, and the exceptional set is thus assumed to be connected with finite length. As applications of the main result, several corollaries are obtained with interesting features.