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Bulletin of the London Mathematical Society Advance Access published online on April 23, 2008
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdn029
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© 2008 London Mathematical Society

Commuting holonomies and rigidity of holomorphic foliations

Hossein Movasati

IMPA, Estrada D. Castorina, 110
Jardim Botânico
Rio de Janeiro, RJ
CEP. 22460-320
Brazil

Isao Nakai

Ochanomizu University
Department of Mathematics
2-1-1 Otsuka, Bunkyo-ku
Tokyo 112-8610
Japan
nakai@math.ocha.ac.jp

Received 15 March 2007. Revision received 8 January 2008.

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 that 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.

2000 Mathematics Subject Classification 57R30, 14D99, 32G34.

The first author is supported by the Japan Society for the Promotion of Sciences.


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