# zbMATH — the first resource for mathematics

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.

##### 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 $$\mathbb{C}$$
##### Keywords:
foliations by curves on $$\mathbb{C} P(3)$$
Full Text:
##### References:
 [1] V. ARNOL’D, Chapitres supplémentaires de la théorie des equations differentielles ordinaires, Ed. MIR, Moscou, 1980. · Zbl 0956.34501 [2] P. BAUM, R. BOTT, Singularities of holomorphic foliations, J. differential geometry, vol. 7 (1972), 279-342. · Zbl 0268.57011 [3] S. S. CHERN, Meromorphic vector fields and characteristic numbers, Scripta Mathematica, vol. XXIX, no 3-4. [4] C. CAMACHO, P. SAD, Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115 (1982), 579-595. · Zbl 0503.32007 [5] P. GRIFFITHS, J. HARRIS, Principles of algebraic geometry, John Wiley, New York, 1978. · Zbl 0408.14001 [6] X. GOMEZ-MONT, L. O-BOBADILLA, Sistemas dinamicos holomorfos en superficies, Aportaciones Matematicas 3, Sociedad Mexicana de Matematica (1989). · Zbl 0855.58049 [7] J. P. JOUANOLOU, Equations de Pfaff algebriques, LNM 708, Springer-Verlag (1979). · Zbl 0477.58002 [8] D. LEHMANN, Residues for invariant submanifolds of foliations with singularities, Annales de l’Institut Fourier, vol. 41, fasc. 1 (1991), 211-258. · Zbl 0727.57024 [9] A. LINS NETO, Algebraic solutions of polynomial differential equations and foliations in dimension two, LNM 1345, Springer-Verlag (1988). · Zbl 0677.58036 [10] A. LINS NETO, 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] J. PALIS, W. de MELO, Geometric theory of dynamical systems, Springer-Verlag (1982). · Zbl 0491.58001 [12] K. SAITO, Quasihomogene isolierte singularitäten von hyperflächen, Inventiones Math., 14 (1971), 123-142. · Zbl 0224.32011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.