Extensions of complex varieties across \(C^ 1\) manifolds (Q1310146)

In 1970, B. Shiffman proved that a closed subset \(E\) in an open set \(\Omega \subset \mathbb{C}^ n\) with zero \((2k-1)\)-dimensional Hausdorff measure does not obstruct complex varieties, in the sense that if \(V\) is a \(k\)-dimensional complex variety in \(\Omega \backslash E\) and if \(\overline V\) is the closure of \(V\) in \(\Omega\), then \(\overline V \cap \Omega\) is also a \(k\)-dimensional complex variety in \(\Omega\). Here, the author considers the following problem: if \(E\) does not obstruct the variety \(V\) topologically (roughly speaking, that means that there are two local components of \(V\) at every point of \(E)\) is it necessary that \(\overline V \cap \Omega\) also be a complex variety? The main result of the paper gives an affirmative answer to this question when \(E\) is a \((2k-1)\)-dimensional \(C^ 1\) manifold.