Meromorphic convexity is defined on complex manifolds to introduce M-manifolds, a class containing Stein and projective manifolds as well as long C^2 without nonconstant holomorphic functions.
Union of holomorphically convex spaces
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abstract
In this short note, we collect some results regarding the Remmert reduction of holomorphically convex space and its application to a variation of the usual union problem. Classically, the union problem asks the following question: is a complex space, which is an increasing union of Stein subspaces $X_1\Subset X_2\Subset\cdots$, a Stein space itself? The variation we are interested in is the following: is a complex space, which is an increasing union of holomorphically convex subspaces $X_1\Subset X_2\Subset\cdots$, holomorphically convex itself? The results presented here are close analogues of (some of) those alredy present in the literature for the Stein case; our aim is only to collect such material for reference, as we consider it well known.
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Meromorphic Convexity on Complex Manifolds
Meromorphic convexity is defined on complex manifolds to introduce M-manifolds, a class containing Stein and projective manifolds as well as long C^2 without nonconstant holomorphic functions.