交换完整性与全纯叶理的刚性
Commuting holonomies and rigidity of holomorphic foliations
摘要：
本文研究了复平面上具有一般非有理首次积分的全纯叶理的变形。考虑首次积分的正则纤维中的两个消失环，它们具有非零的自相交，且消失路径仅在它们的起点相交。证明了如果这类消失圈的变形完整性可交换，则变形叶理也有一个首次积分。我们的结果推广了Ilyashenko关于具有持久中心奇点的全纯叶理的刚性的一个类似结果。证明的主要工具是Picard-Lefschetz理论和这种变形的迭代积分理论。
英文摘要：
In this article we study deformations of a holomorphic foliation with a generic non-rational first integral in the complex plane. We consider two vanishing cycles in a regular fiber of the first integral with a non-zero self intersection and with vanishing paths which intersect each other only at their start points. It is proved that if the deformed holonomies of such vanishing cycles commute then the deformed foliation has also a first integral. Our result generalizes a similar result of Ilyashenko on the rigidity of holomorphic foliations with a persistent center singularity. The main tools of the proof are Picard-Lefschetz theory and the theory of iterated integrals for such deformations.
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