An envelope of holomorphy for certain normal complex spaces (Q1057399)

The three most general categories in which the existence of an envelope of holomorphy is guaranteed are the category of unbranched Riemann domains over \({\mathbb{C}}^ n\), the category of branched Riemann domains over \({\mathbb{C}}^ n\), and the category of holomorphically convex complex spaces. - Branched Riemann domains over \({\mathbb{C}}^ n\) are holomorphically spreadable. There are many important non-spreadable spaces, however, which also have an envelope of holomorphy. The simplest example is obtained by blowing up the origin in \({\mathbb{C}}^ 2\); another such space is a counterexample of Skoda to the Serre problem. The purpose of this paper is to show that every connected normal complex space X whose separation relation \(R^ X\) is locally semiproper has an envelope of holomorphy. The equivalence relation \(R^ X\) is given by identifying those points of X which cannot be separated by global holomorphic functions. Since \(R^ X\) is locally proper for a holomorphically spreadable space X, the category of normal spaces considered in this paper contains the subcategory of branched Riemann domains over \({\mathbb{C}}^ n\). Two interesting properties of the envelope of holomorphy H(X) of connected normal spaces X with a locally semiproper separation relation are that dim H(X)\(\leq \dim X\) holds and that H(X) is holomorphically spreadable, even though X need not be spreadable. Examples for which dim H(X)\(<\dim X\) is true are also mentioned.