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 Mosc. Math. J., 2011, Volume 11, Number 3, Pages 521–530 (Mi mmj430)

Total rigidity of generic quadratic vector fields

Yu. Ilyashenkoabcd, V. Moldavskisde

a Steklov Math. Institute, Moscow, Russia
b Moscow Independent University
c Moscow State University
d Cornell University, US
e London, UK

Abstract: We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb CP^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the and in $\mathbb CP^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a finite number of foliations of the same class, modulo affine equivalence. This property is called total rigidity. A recent result of Lins Neto implies that the finite number above does not exceed 240.
This is the first of the two closely related papers. It deals with the rigidity properties of quadratic foliations, whilst the second one studies the foliations of higher degree.

Key words and phrases: foliations, topological equivalence, rigidity.

DOI: https://doi.org/10.17323/1609-4514-2011-11-3-521-530

Full text: http://www.ams.org/.../abst11-3-2011.html
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MSC: 37F75
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Citation: Yu. Ilyashenko, V. Moldavskis, “Total rigidity of generic quadratic vector fields”, Mosc. Math. J., 11:3 (2011), 521–530

Citation in format AMSBIB
\Bibitem{IlyMol11} \by Yu.~Ilyashenko, V.~Moldavskis \paper Total rigidity of generic quadratic vector fields \jour Mosc. Math.~J. \yr 2011 \vol 11 \issue 3 \pages 521--530 \mathnet{http://mi.mathnet.ru/mmj430} \crossref{https://doi.org/10.17323/1609-4514-2011-11-3-521-530} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2894428} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000300365900006} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Calsamiglia G., Deroin B., Frankel S., Guillot A., “Singular Sets of Holonomy Maps for Algebraic Foliations”, J. Eur. Math. Soc., 15:3 (2013), 1067–1099
2. Ramirez V., “The Utmost Rigidity Property For Quadratic Foliations on P-2 With An Invariant Line”, Bol. Soc. Mat. Mex., 23:2 (2017), 759–813
3. Ramirez V., “Twin Vector Fields and Independence of Spectra For Quadratic Vector Fields”, J. Dyn. Control Syst., 23:3 (2017), 623–633