On Lins Neto’s examples of algebraic foliations. (Sur les exemples de Lins Neto de feuilletages algébriques.) (French. Abridged English version) Zbl 1004.37029

Let \(X_0\) and \(X_\infty\) be holomorphic vector fields on \(\mathbb{C}^3\) defined by \[ X_0=(-3z_1^2+ z^2_2+2z_1z_3) \partial/\partial z_1+2z_2(-3z_1+2z_3) \partial/ \partial z_2+2z_3 (3z_1-z_3)\partial/ \partial z_3 \] and \[ X_\infty= 2z_2(-z_1+z_3) \partial/ \partial z_1+(3z_1^2- z^2_2)\partial/ \partial z_2+2z_3z_2 \partial/ \partial z_3, \] where \((z_1,z_2,z_3)\) denotes the coordinates on \(\mathbb{C}^3\). Set \(X_\alpha= X_0+\alpha X_\infty\) for \(\alpha\in \mathbb{C}\). Let \(\Lambda\) be the lattice in \(\mathbb{C}\), generated by 1 and \(\omega\), where \(\omega\) is a primitive 6-th root of unity. The author proves, among other things, that \(X_\alpha\) is completely integrable if and only if \((\alpha-1)/2= \lambda_1/ \lambda_2\) for some \(\lambda_1\), \(\lambda_2 \in\Lambda\). The author asserts that the family \(\{X_\alpha\}\) gives a counterexample to the conjecture of Painlevé, which says that every polynomial vector field on \(\mathbb{C}^n\) with global meromorphic solutions is completely integrable.


37F75 Dynamical aspects of holomorphic foliations and vector fields
32S65 Singularities of holomorphic vector fields and foliations
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57R30 Foliations in differential topology; geometric theory
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[1] Ercolani, N.; Siggia, E. D., Painlevé property and integrability, (What is Integrability? (1991), Springer: Springer Berlin), 63-72 · Zbl 0733.34008
[2] Ghys, É.; Rebelo, J.-C., Singularités des flots holomorphes, II, Ann. Inst. Fourier (Grenoble), 47, 4, 1117-1174 (1997), Erratum op. cit. 50 (3) (2000) 1019-1020 · Zbl 0938.32019
[3] A. Lins Neto, Some examples for Poincaré and Painlevé problems, Prépublication, 2000; A. Lins Neto, Some examples for Poincaré and Painlevé problems, Prépublication, 2000
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