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 Mosc. Math. J., 2012, Volume 12, Number 4, Pages 825–862 (Mi mmj484)

Thom's problem for degenerated singular points of holomorphic foliations in the plane

L. Ortiz-Bobadillaa, E. Rosales-Gonzáleza, S. M. Voroninb

a Instituto de Matemáticas, Universidad Nacional Autonoma de México
b Departament of Mathematics, Chelyabinsk State University

Abstract: Let $\mathcal{V}_n$ be the class of germs of holomorphic non-dicritic vector fields in $(\mathbb{C}^2,0)$ with vanishing $(n-1)$-jet at the origin, $n\geq2$, and non-vanishing $n$-jet. In the present work the formal normal form (under the strict orbital classification) of generic germs in a subclass $\mathcal{V}_n^o$ of $\mathcal{V}_n$ is given. Any such normal form is given as the sum of three terms: a “principal” generic homogeneous term, $\mathbf{v}_o\in\mathcal{V}_n$, a “hamiltonian” term, $\mathbf{v}_c$ (given by a hamiltonian polynomial vector field) and a “radial” term.
For any generic germ $\mathbf{v}\in\mathcal{V}_n^o$ we define the triplet $i_\mathbf{v}= (\mathbf{v}_o, \mathbf{v}_c,[G_{\mathbf{v}}])$, where $\mathbf{v}_o$ and $\mathbf{v}_c$ denote the principal and hamiltonian terms of its corresponding formal normal form, and $[G_{\mathbf{v}}]$ denotes the class of strict analytic conjugacy of its projective (hidden or vanishing) monodromy group. We prove that the terms appearing in $i_{\mathbf{v}}$ are Thom's invariants of the strict analytical orbital classification of generic germs in $\mathcal{V}_n^o$: two generic germs $\mathbf{v}$ and $\tilde{\mathbf{v}}$ in $\mathcal{V}_n^o$ are strictly orbitally analytically equivalent if and only if $i_{\mathbf{v}}= i_{\tilde{\mathbf{v}}}$. Moreover, any triplet satisfying some natural conditions of concordance can be realized as invariant of a generic germ of $\mathcal{V}_n^o$.

Key words and phrases: Non-dicritic foliations, non-dicritic vector fields, formal normal forms, analytic invariants, monodromy group.

DOI: https://doi.org/10.17323/1609-4514-2012-12-4-825-862

Full text: http://www.mathjournals.org/.../2012-012-004-010.html
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Bibliographic databases:

MSC: Primary 32S65, 37F75; Secondary 32S70, 32S05, 32S30, 34A25, 34C20, 57R30
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Citation: L. Ortiz-Bobadilla, E. Rosales-González, S. M. Voronin, “Thom's problem for degenerated singular points of holomorphic foliations in the plane”, Mosc. Math. J., 12:4 (2012), 825–862

Citation in format AMSBIB
\Bibitem{OrtRosVor12} \by L.~Ortiz-Bobadilla, E.~Rosales-Gonz\'alez, S.~M.~Voronin \paper Thom's problem for degenerated singular points of holomorphic foliations in the plane \jour Mosc. Math.~J. \yr 2012 \vol 12 \issue 4 \pages 825--862 \mathnet{http://mi.mathnet.ru/mmj484} \crossref{https://doi.org/10.17323/1609-4514-2012-12-4-825-862} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3076858} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000314341500010} 

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This publication is cited in the following articles:
1. V. Ramirez, “An example of a non-algebraizable singularity of a holomorphic foliation”, Enseign. Math., 62:1-2 (2016), 7–14
2. J. Angelica Jaurez-Rosas, “Real-formal orbital rigidity for germs of real analytic vector fields on the real plane”, J. Dyn. Control Syst., 23:1 (2017), 89–109
3. G. Calsamiglia, Y. Genzmer, “Classification of regular dicritical foliations”, Ergod. Theory Dyn. Syst., 37:5 (2017), 1443–1479
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