Convexity properties of analytic complements in Stein spaces (Q1378958)

From the Introduction: ``It is well known that geometric convexity properties of complex manifolds or, more generally, complex spaces, imply strong analytic consequences. For the case of geometric 1-completeness this is the heart of the solution of the Levi problem together with Theorem B of Cartan and Serre; more generally, for \(q\)-completeness this follows from the fundamental theory of Andreotti and Grauert. Therefore, it is important to have criteria that give geometric convexity of suitable classes of complex spaces. The most general fact in this respect is probably the results from T. Ohsawa (1984) and J. P. Demailly (1990) namely, any complex space \(X\) of dimension \(n\) without any compact irreducible \(n\)-dimensional component is \(n\)-complete. For further results more specific hypotheses have to be made. An important situation that appears very often in complex analysis is the following: A complex space \(X\) is given together with a certain complex analytic subvariety \(A\subset X\), and one wants to study properties of the complement \(X':=X\setminus A\). In the light of the aforementioned facts, it can then be important to know how the convexity properties of \(X\) and the nature of \(A\) influence the convexity of the complement \(X'\). Many more specific questions of this sort are still open. In this article, we want to consider the case in which \(X\) is known to be Stein, or, in other words, 1-complete, and \(A\) has strictly positive dimension at any of its points. In terms of \(q\)-convexity (resp. \(q\)-completeness) with corners, this question has been studied by M. Peternell (1986). The main purpose of this article now is to show that in the situation of the corollary \(X'\) is at least always \((n-1)\)-complete if \(q<n\). In fact, more generally, we show the following theorem. Theorem. Let \(X\) be a Stein space of pure dimension \(n\geq 2\) and \(A\subset X\) a (closed) complex subvariety such that for any point \(x\in A\) and any local irreducible component \(X_{i,x}\) of \(X\) at \(x\), the point \(x\) is not isolated in \(A\cap X_{i,x}\). Then \(X\subset A\) is a union of \((n-1)\) open Stein subspaces. In particular, \(X\setminus A\) is \((n-1)\)-complete''.