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

Guillot, Adolfo

C. R., Math., Acad. Sci. Paris 334, No. 9, 747-750 (2002).

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.

Reviewer: A.Morimoto (Nagoya)

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

Keywords:

complete integrability; holomorphic vector fields

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