Schlicht Envelopes of Holomorphy and Foliations by Lines

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.