Abstract

The following result is proved. Let \(M\) be a complex space and \(\mathbb{B}^n\) the open unit ball in \(\mathbb{C}^n\). Let \(f: \mathbb{B}^n \rightarrow M\) be a mapping such that \(f\) is holomorphic on the intersection of \(\mathbb{B}^n\) with every complex line passing through the origin, and \(f\) is \(C^\infty\) smooth in a neighborhood of the origin. Then there exists a pluripolar subset \(S\) of \(\mathbb{P}^{n-1}(\mathbb{C})\) such that \(f\) is holomorphic in a neighborhood of \(\mathbb{B}^n - \bigcup\{l: l \in S \}\). If \(M\) is a holomorphically convex compact Kähler manifold, in the above case, then \(f\) is meromorphic in \(\mathbb{B}^n\).