On algebraic sets invariant by one-dimensional foliations on \(\mathbb{C} P(3)\). (English) Zbl 0770.57016
We consider the problem of extending the result of J.-P. Jouanolou on the density of singular holomorphic foliations on \(\mathbb{C} P(2)\) without algebraic solutions to the case of foliations by curves on \(\mathbb{C} P(3)\). We give an example of a foliation on \(\mathbb{C} P(3)\) with no invariant algebraic set (curve or surface) and prove that a dense set of foliations admits no invariant algebraic set.
Reviewer: Marcio G.Soares (Belo Horizonte)
MSC:
| 57R30 | Foliations in differential topology; geometric theory |
| 34M99 | Ordinary differential equations in the complex domain |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
References:
| [1] | [A] , Chapitres Supplémentaires de la Théorie des Equations Differentielles Ordinaires, Ed. MIR, Moscou, 1980. · Zbl 0956.34501 |
| [2] | [BB] , , Singularities of Holomorphic Foliations, J. Differential Geometry, vol. 7 (1972), 279-342. · Zbl 0268.57011 |
| [3] | [C] , Meromorphic Vector Fields and Characteristic Numbers, Scripta Mathematica, vol. XXIX, no 3-4. |
| [4] | [CS] , , Invariant Varieties through Singularities of Holomorphic Vector Fields, Annals of Math., 115 (1982), 579-595. · Zbl 0503.32007 |
| [5] | [GH] , , Principles of Algebraic Geometry, John Wiley, New York, 1978. · Zbl 0408.14001 |
| [6] | [GM-OB] , , Sistemas Dinamicos Holomorfos en Superficies, Aportaciones Matematicas 3, Sociedad Mexicana de Matematica (1989). · Zbl 0855.58049 |
| [7] | [J] , Equations de Pfaff Algebriques, LNM 708, Springer-Verlag (1979). · Zbl 0477.58002 |
| [8] | [L] , Residues for Invariant Submanifolds of Foliations with Singularities, Annales de l’Institut Fourier, vol. 41, fasc. 1 (1991), 211-258. · Zbl 0727.57024 |
| [9] | [LN1] , Algebraic Solutions of Polynomial Differential Equations and Foliations in Dimension Two, LNM 1345, Springer-Verlag (1988). · Zbl 0677.58036 |
| [10] | [LN2], Complex Codimension One Foliations leaving a Compact Submanifold Invariant, Dynamical Systems and Bifurcation Theory, Pitman Research Notes in Mathematics Series, vol. 160 (1987). · Zbl 0647.57017 |
| [11] | [PM] , , Geometric Theory of Dynamical Systems, Springer-Verlag (1982). · Zbl 0491.58001 |
| [12] | [S] , Quasihomogene Isolierte Singularitäten von Hyperflächen, Inventiones Math., 14 (1971), 123-142. · Zbl 0224.32011 |
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