Schlicht envelopes of holomorphy and foliations by lines

Details Schlicht envelopes of holomorphy and foliations by lines Schlicht envelopes of holomorphy and foliations by lines

2008-02-06

arxiv.org/abs/0802.0727v3

Mathematics

Complex Variables

Version 2: Terminology revised and title changed in view of information about the origins of what we now call "the schlichtnes

Scientific paper

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.

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