Several complex variables and analytic spaces Holomorphic convexity and pseudoconvexity Problem

Levi problem

Builds on Pseudoconvexity Domain of holomorphy

Problem

Let $n \ge 2$ and let $\Omega \subseteq \C^n$ be a domain. Every domain of holomorphy is pseudoconvex. Is every pseudoconvex domain $\Omega \subseteq \C^n$ a domain of holomorphy?

Remarks

The answer is yes. Kiyoshi Oka proved it for $n = 2$ in 1942. For all $n$ it was proved in the early 1950s, independently by Oka, by Hans-Joachim Bremermann and by François Norguet. Hans Grauert solved the analogue for complex manifolds in 1958: a complex manifold with a smooth strictly plurisubharmonic exhaustion function is a Stein manifold. A modern proof solves the equation $\bar\partial u = f$ with Hörmander's $L^2$ estimates for weights built from a plurisubharmonic exhaustion function. The problem has no content for $n = 1$ , where every domain is both pseudoconvex and a domain of holomorphy.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Tag	`47CVS9RJ9D`
URI	`m2://47CVS9RJ9D`
Kind	problem
Writing	W0 stub
MSC	32E40, 32T05
Status	This copy of the site has no build data. Run `m2 status 47CVS9RJ9D`.
About	Pseudoconvexity, Domain of holomorphy
Uses	Pseudoconvexity, Domain of holomorphy

Levi problem

Problem

Remarks

Reading path

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