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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.

MSC:

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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References:

[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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