Levi Problem in Complex Manifolds

TL;DR

This paper investigates the conditions under which pseudoconvex open sets in complex manifolds are Stein, introducing Levi-currents as a key tool to analyze obstructions and providing new criteria for Steinness.

Contribution

It introduces new criteria for Steinness of pseudoconvex sets using Levi-currents and explores their role in the Levi problem within complex manifolds.

Findings

Levi-currents characterize obstructions to Steinness.

Constructs bounded strictly plurisubharmonic exhaustion functions under geometric conditions.

Relates foliation dynamics to solutions of the Levi problem.

Abstract

Let U be a pseudoconvex open set in a complex manifold M. When is U a Stein manifold? There are classical counter examples due to Grauert, even when U has real-analytic boundary or has strictly pseudoconvex points. We give new criteria for the Steinness of U and we analyze the obstructions. The main tool is the notion of Levi-currents. They are positive $\ddbar$-closed currents T of bidimension (1,1) and of mass 1 directed by the directions where all continuous psh functions in $U$ have vanishing Levi-form. The extremal ones, are supported on the sets where all continuous psh functions are constant. We also construct under geometric conditions, bounded strictly psh exhaustion functions, and hence we obtain Donnelly- Fefferman weights. To any infinitesimally homogeneous manifold, we associate a foliation. The dynamics of the foliation determines the solution of the Levi-problem. Some of the results can be extended to the context of pseudoconvexity with respect to a Pfaff-system.