Abstract
Levi Problem: Complement of a closed subspace in a Stein space and its applications
Highlights
We consider the following Levi Problem in this paper
If the boundary X −Y is a closed subspace in X such that locally at every point in X −Y, X −Y is defined by one holomorphic function, Y is Stein
For every point p0 ∈ A − A ∩ Y = A ∩ (X − Y ), there is a hypersurface Zh defined by a holomorphic function h in X such that Z = Y ∩ Zh is locally Stein in Zh and there is a holomorphic function G on a Stein open subset W ⊃ Z ∩ A with polynomial growth on W such that G is not bounded on any sequence in W with the accumulation point p0
Summary
We consider the following Levi Problem in this paper. The detailed discussions and historic developments of the Levi Problem can be found in many literatures, e.g., [8, 29, 31]. Andreotti and Narasimhan proved that if X is a Stein space with isolated singularities, a locally Stein open subset Y in X is Stein [1] Their proof heavily relies on the fact that − log d is a pseudoconvex function, where d is the distance function ([1, 29]). If the boundary X −Y is a closed subspace in X such that locally at every point in X −Y , X −Y is defined by one holomorphic function, Y is Stein. Since the Levi problem for curves and surfaces in Theorem 2 has an affirmative answer [26], we assume that the complex space X is of dimension at least 3
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