Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Liouvillian integration and Bernoulli foliations


Authors: D. Cerveau and P. Sad
Journal: Trans. Amer. Math. Soc. 350 (1998), 3065-3081
MSC (1991): Primary 32L30; Secondary 58F18
DOI: https://doi.org/10.1090/S0002-9947-98-01780-2
MathSciNet review: 1390971
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under ``generic'' extra conditions, it is proven that these foliations can be defined by Bernoulli equations.


References [Enhancements On Off] (What's this?)

  • 1. C. Camacho, A. Lins Neto and P. Sad, Foliations with Algebraic Limit Sets, Ann. of Math. 136 (1992). MR 93i:32035
  • 2. C. Camacho and P. Sad, Invariant Varieties Through Singularities of Holomorphic Vector Fields, Ann. of Math. 115 (1982). MR 83m:58062
  • 3. D. Cerveau and A. Lins Neto, Holomorphic Foliations in $\mathbf{C}P(2)$ having an Invariant Algebraic Curve, Ann. Inst. Fourier 41 (1991). MR 93b:32050
  • 4. D. Cerveau et J-F. Mattei, Formes Intégrables Holomorphes Singulières, Astérisque 97, Soc. Math. France, 1982. MR 86f:58006
  • 5. D. Cerveau et R. Moussu, Groupes d'Automorphismes de $(\mathbf{C},0)$ et Équations Différentielles $ydy+\cdots =0$, Bull. Soc. Math. France. 116 (1988). MR 90m:58192
  • 6. P. Deligne, Le Groupe Foundamental du Complément d'une Courbe Plane n'ayant que des Points Doubles Ordinaires est Abélian, Sem. Bourbaki, Lecture Notes in Math. 842, Springer, 1981. MR 83f:14026
  • 7. I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. MR 20:177
  • 8. A. Lins Neto, Construction of Singular Holomorphic Vector Fields in Dimension Two, J. Differential Geometry 26 (1987). MR 88f:32047
  • 9. J-F. Mattei et R. Moussu, Holonomie et Intégrales Premières, Ann. Sci. Ecole Norm. Sup. 13 (1980). MR 83b:58005
  • 10. M. Prelle and M. Singer, Elementary First Integrals of Differential Equations, Trans. Amer. Math. Soc. 279 (1983), 215-229. MR 85d:12008
  • 11. B. Scárdua, Transversely Affine Foliations on Complex Projective Spaces, IMPA, Thesis.
  • 12. B. Seke, Sur les Structures Transversalement Affines des Feuilletages de Codimension Un, Ann. Inst. Fourier 30 (1980). MR 82b:57023
  • 13. M. Singer, Liouvillian First Integrals of Differential Equations, Trans. Amer. Math. Soc. 333 (1992), 673-688. MR 92m:12014
  • 14. J. Tits, Free Subgroups in Linear Groups, J. Algebra 20 (1972). MR 44:4105

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32L30, 58F18

Retrieve articles in all journals with MSC (1991): 32L30, 58F18


Additional Information

D. Cerveau
Affiliation: Université de Rennes I - IRMAR, Campus de Beaulieu - 35042, Rennes, France

P. Sad
Affiliation: Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, CEP 22460-320 Rio de Janeiro, Brazil

DOI: https://doi.org/10.1090/S0002-9947-98-01780-2
Received by editor(s): December 7, 1994
Received by editor(s) in revised form: November 28, 1995
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society