2011/12/29 by Youssef Alaoui, Alaoui, Youssef
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1112.6292
openalex publication_date 2011/12/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
We show that if X is a Stein space and, if Ω⊂ X is exhaustable by a sequence Ω1 ⊂ Ω2 ⊂ … ⊂ Ωn ⊂ … of open Stein subsets of X, then Ω is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for X=ℂn and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When X has dimension 2, we prove that the same result follows if we assume only that Ω⊂ ⊂ X is a domain of holomorphy in a Stein normal space. It is known, however, that if X is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets X1 ⊂ X2 ⊂ ⋯ ⊂ Xn ⊂ ⋯, it does not follow in general that X is holomorphically-convex or holomorphically-separate (even if X has no singularities). One can even obtain 2-dimensional complex manifolds on which all holomorphic functions are constant.