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 Trudy Mat. Inst. Steklova, 2007, Volume 259, Pages 64–76 (Mi tm569)

This article is cited in 1 scientific paper (total in 1 paper)

Total Rigidity of Polynomial Foliations on the Complex Projective Plane

Yu. S. Ilyashenkoabcd

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute, Russian Academy of Sciences
c Independent University of Moscow
d Cornell University

Abstract: Polynomial foliations of the complex plane are topologically rigid. Roughly speaking, this means that the topological equivalence of two foliations implies their affine equivalence. There exist various nonequivalent formalizations of the notion of topological rigidity. Generic polynomial foliations of fixed degree have the so-called property of absolute rigidity, which is the weakest form of topological rigidity. This property was discovered by the author more than 30 years ago. The genericity conditions imposed at that time were very restrictive. Since then, this topic has been studied by Shcherbakov, Gómez-Mont, Nakai, Lins Neto–Sad–Scárdua, Loray–Rebelo, and others. They relaxed the genericity conditions and increased the dimension. The main conjecture in this field states that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations. The main result of this paper is weaker than this conjecture but also makes it possible to compare topological types of distant foliations.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2007, 259, 60–72

Bibliographic databases:

UDC: 517.927.7
Received in November 2006

Citation: Yu. S. Ilyashenko, “Total Rigidity of Polynomial Foliations on the Complex Projective Plane”, Analysis and singularities. Part 2, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 259, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 64–76; Proc. Steklov Inst. Math., 259 (2007), 60–72

Citation in format AMSBIB
\Bibitem{Ily07} \by Yu.~S.~Ilyashenko \paper Total Rigidity of Polynomial Foliations on the Complex Projective Plane \inbook Analysis and singularities. Part~2 \bookinfo Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday \serial Trudy Mat. Inst. Steklova \yr 2007 \vol 259 \pages 64--76 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm569} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2433677} \zmath{https://zbmath.org/?q=an:1156.32302} \elib{https://elibrary.ru/item.asp?id=9572728} \transl \jour Proc. Steklov Inst. Math. \yr 2007 \vol 259 \pages 60--72 \crossref{https://doi.org/10.1134/S0081543807040050} \elib{https://elibrary.ru/item.asp?id=13549998} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38849204350} 

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This publication is cited in the following articles:
1. Yu. Ilyashenko, V. Moldavskis, “Total rigidity of generic quadratic vector fields”, Mosc. Math. J., 11:3 (2011), 521–530
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