On algebraic sets invariant by one-dimensional foliations on 𝐂P(3)
Annales de l'Institut Fourier, Volume 43 (1993) no. 1, p. 143-162

We consider the problem of extending the result of J.-P. Jouanolou on the density of singular holomorphic foliations on CP(2) without algebraic solutions to the case of foliations by curves on CP(3). We give an example of a foliation on CP(3) with no invariant algebraic set (curve or surface) and prove that a dense set of foliations admits no invariant algebraic set.

On considĂšre le problĂšme d’étendre le rĂ©sultat de J.-P. Jouanolou Ă  la densitĂ© des feuilletages holomorphes singuliers dans CP(2), sans solution algĂ©brique, au cas des feuilletages par des courbes dans CP(3). On donne un exemple de feuilletage dans CP(3) sans ensemble algĂ©brique invariant (courbe ou surface) et on montre qu’un ensemble dense de feuilletages n’admet pas d’ensemble algĂ©brique invariant.

@article{AIF_1993__43_1_143_0,
     author = {Soares, Marcio G.},
     title = {On algebraic sets invariant by one-dimensional foliations on ${\bf C}P(3)$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {1},
     year = {1993},
     pages = {143-162},
     doi = {10.5802/aif.1325},
     zbl = {0770.57016},
     mrnumber = {94b:32057},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_1_143_0}
}
Soares, Marcio G. On algebraic sets invariant by one-dimensional foliations on ${\bf C}P(3)$. Annales de l'Institut Fourier, Volume 43 (1993) no. 1, pp. 143-162. doi : 10.5802/aif.1325. http://www.numdam.org/item/AIF_1993__43_1_143_0/

[A] V. Arnol'D, Chapitres Supplémentaires de la Théorie des Equations Differentielles Ordinaires, Ed. MIR, Moscou, 1980. | MR 626685 | MR 83a:34003 | Zbl 0455.34001 | Zbl 0956.34501

[BB] P. Baum, R. Bott, Singularities of Holomorphic Foliations, J. Differential Geometry, vol. 7 (1972), 279-342. | MR 377923 | MR 51 #14092 | Zbl 0268.57011

[C] S. S. Chern, Meromorphic Vector Fields and Characteristic Numbers, Scripta Mathematica, vol. XXIX, no 3-4. | MR 412486 | Zbl 0265.32013

[CS] C. Camacho, P. Sad, Invariant Varieties through Singularities of Holomorphic Vector Fields, Annals of Math., 115 (1982), 579-595. | MR 657239 | MR 83m:58062 | Zbl 0503.32007

[GH] P. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley, New York, 1978. | MR 507725 | MR 80b:14001 | Zbl 0408.14001

[GM-OB] X. Gomez-Mont, L. O-Bobadilla, Sistemas Dinamicos Holomorfos en Superficies, Aportaciones Matematicas 3, Sociedad Mexicana de Matematica (1989). | MR 1304495 | MR 95j:32046 | Zbl 0855.58049

[J] J. P. Jouanolou, Equations de Pfaff Algebriques, LNM 708, Springer-Verlag (1979). | MR 537038 | MR 81k:14008 | Zbl 0477.58002

[L] D. Lehmann, Residues for Invariant Submanifolds of Foliations with Singularities, Annales de l'Institut Fourier, vol. 41, fasc. 1 (1991), 211-258. | Numdam | MR 1112198 | MR 92k:57054 | Zbl 0727.57024

[LN1] A. Lins Neto, Algebraic Solutions of Polynomial Differential Equations and Foliations in Dimension Two, LNM 1345, Springer-Verlag (1988). | MR 980960 | MR 90c:58142 | Zbl 0677.58036

[LN2]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). | MR 88m:57036 | Zbl 0647.57017

[PM] J. Palis, W. De Melo, Geometric Theory of Dynamical Systems, Springer-Verlag (1982). | Zbl 0491.58001

[S] K. Saito, Quasihomogene Isolierte SingularitÀten von HyperflÀchen, Inventiones Math., 14 (1971), 123-142. | MR 45 #3767 | Zbl 0224.32011