全纯的Schlicht包络和线的叶理
Schlicht envelopes of holomorphy and foliations by lines
摘要：
给定复流形X中的一个区域Y，判定Y在X中是否有一个全纯的schlicht全纯包，以及如果有，如何描述这个包，是一个没有一般解的难题。本文的目的是解决这个问题的帮助下，光滑的一维叶理F的X没有紧凑的叶子。我们称X中的整环Y是关于F的区间整环，如果Y与F的每个叶在非空连通集中相交。证明了若X是Stein且F满足一个新的性质称为拟全纯性，则X中的每个区间整环都有一个全纯的Schlicht全纯包，它也是一个区间整环.这个结果是20世纪80年代中期一个著名引理的推广和全球版本。我们用充分条件、例子和反例说明了拟全纯性的概念，并给出了一些应用，特别是一个很少研究的域的边界正则性，称为局部schlichtness。
英文摘要：
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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