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Moscow Mathematical Journal

Volume 12, Issue 4, October–December 2012  pp. 825–862.

Thom's Problem for Degenerate Singular Points of Holomorphic Foliations in the Plane

Authors L. Ortiz-Bobadilla (1), E. Rosales-González (1), and S. Voronin (2)
Author institution: Instituto de Matemáticas, Universidad Nacional Autonoma de México
Departament of Mathematics, Chelyabinsk State University

Summary:  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, $\boldsymbol{v}_o\in\mathcal{V}_n$, a ``hamiltonian'' term, $\boldsymbol{v}_{c} $ (given by a hamiltonian polynomial vector field) and a ``radial'' term. For any generic germ $\boldsymbol{v}\in\mathcal{V}_n^o$ we define the triplet $i_{\boldsymbol{v}}= (\boldsymbol{v}_o, \boldsymbol{v}_{c},[G_{\boldsymbol{v}}])$, where $\boldsymbol{v}_o$ and $\boldsymbol{v}_{c}$ denote the principal and hamiltonian terms of its corresponding formal normal form, and $[G_{\boldsymbol{v}}]$ denotes the class of strict analytic conjugacy of its projective (hidden or vanishing) monodromy group. We prove that the terms appearing in $i_{\boldsymbol{v}}$ are Thom's invariants of the strict analytical orbital classification of generic germs in $\mathcal{V}_n^o$: two generic germs $\boldsymbol{v}$ and $\tilde{\boldsymbol{v}}$ in $\mathcal{V}_n^o$ are strictly orbitally analytically equivalent if and only if $i_{\boldsymbol{v}}= i_{\tilde{\boldsymbol{v}}}$. Moreover, any triplet satisfying some natural conditions of concordance can be realized as invariant of a generic germ of $\mathcal{V}_n^o$.

2010 Mathematics Subject Classification. Primary: 32S65, 37F75; Secondary: 32S70, 32S05, 32S30, 34A25, 34C20, 57R30
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