Union of holomorphically convex spaces

TL;DR

This paper explores whether increasing unions of holomorphically convex spaces remain holomorphically convex, extending classical results from Stein spaces and providing a collection of known results for reference.

Contribution

It collects and discusses results on the union problem for holomorphically convex spaces, extending classical Stein space results to a broader context.

Findings

Union of holomorphically convex spaces can be holomorphically convex under certain conditions

Results are analogous to those known for Stein spaces

Provides a reference collection of known results

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.