Generic pseudogroups on $( \mathbb{C}, 0)$ and the topology of leaves. (August 2013)

• Record Type: Journal Article
• Authors: Mattei, J.-F.; Rebelo, J. C.; Reis, H.
• Is Part Of: Compositio mathematica. Volume 149:Number 8(2013)
• Page Start: 1401
• Page End: 1430
• Publication Date: 2013-08
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X13007161
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg900t1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$0\in \mathbb{C} $]]></tex-math></alternatives></inline-formula> is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg900jm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\{ {y}^{2} - {x}^{2n+ 1} = 0\} $]]></tex-math></alternatives></inline-formula>, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.</p> </abstract>