Schlicht envelopes of holomorphy and foliations by lines

TL;DR

This paper investigates conditions under which domains in complex manifolds have schlicht envelopes of holomorphy, introducing quasiholomorphic foliations to generalize classical results and explore boundary regularity.

Contribution

It introduces the concept of quasiholomorphicity for foliations and proves that interval domains in Stein manifolds with such foliations have schlicht envelopes, extending classical lemmas.

Findings

Quasiholomorphic foliations ensure schlicht envelopes for interval domains.

Stein manifolds with quasiholomorphic foliations generalize classical results.

Applications include insights into boundary regularity and local schlichtness.

Abstract

Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope of holomorphy, which is also an interval domain. This result is a generalization and a global version of a well-known lemma from the mid-1980s. We illustrate the notion of quasiholomorphicity with sufficient conditions, examples, and counterexamples, and present some applications, in particular to a little-studied boundary regularity property of domains called local schlichtness.