Abstract
The main objective of this article is to study the topology of the fibers of a generic rational function of the type\(\frac{{F^p }}{{G^q }}\) in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle δ generates the first homology group of a generic fiber of\(\frac{{F^p }}{{G^q }}\). In particular, we will prove that for any two Lefschetz vanishing cyclesδ 0 andδ 1 in a regular compact fiber of\(\frac{{F^p }}{{G^q }}\), there exists a mondromyh such thath(δ 0)=±δ 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Abraham and J. Robbin, Transversal Mappings and flows, W. A. Benjamin, Inc. New York, (1967).
V.I. Arnold, S.M. Gusein-zade and A.N. Varchenko, Singularities of Differential Maps, V. II, Birkhäuser, (1988).
D. Chéniot,Vanishing Cycles in a Pencil of Hyperplane Sections of a Nonsingular Quasi-projective Variety, Proc. London Math. Soc.72(3) (1996), 515–544.
D. Cerveau and A. Lins Neto,Irreducible Components of the Space of Holomorphic Foliation of Degree 2 inℂP(N), N≥3, Annals of mathematics,143 (1996), 577–612.
P. Deligne, Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Mathematics, 163, Springer-Verlag, (1970).
A. Dimca, Singularities and Topology of Hypersurfaces, Univeritext, Springer-Verlag, New York Inc. (1992).
C. Ehresmann,Sur l'espaces fibrés différentiables, C.R. Acad. Sci. Paris224 (1947), 1611–1612.
H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, Birkhäuser, DMV Seminar Band20 (1992).
Yu. S. Ilyashenko, The Origin of Limit Cycles Under perturbation of Equation\(\frac{{dw}}{{dz}} = - \frac{{Rz}}{{Rw}}\), WhereR(z, w) Is a Polynomial, Math. USSR, Sbornik,7 (1969), No. 3.
G. R. Kempf, Algebraic Varities, London Mathematical Society, Lecture Note series 172, Cambridge Univercity Press.
K. Lamotke, The Topology of Complex Projective Varieties After S. Lefschetz, Topology, Vol20 (1981).
S. Lefschetz, L'Analysis Situs et la géométrie Algébrique Paris, Gauthier-Villars, (1924).
A. Lins Neto and B. A. Scárdua, Foleações Algébricas Complexas, 21° Colóquio Brasileiro de Matemática.
W.S. Massey, A Basic Course in Algebraic Topology, (1991), Springer-Verlag, New York.
J. Milnor, Morse theory, Princeton Press, Princeton, NJ, (1963).
E. Paul, Cycles Évanescents D'une Fonction de Liouville de Type\(f_1^{\lambda _1 } \cdots f_p^{\lambda p}\), Ann. Inst. Fourier, Grenoble,45(1) (1995), 31–63.
I. R. Shafarevich, Basic Algebraic Geometry, V. I, Springer-Verlag, Berlin Heidelberg, (1977, 1994).
R. Speiser, Transversality Theorems For Families of Maps, Lecture Notes in Mathematics, 1311, Algebraic Geometry Sundance, (1986).
N. Steenrod, The homology of Fiber Bundles, Princeton University press (1951).
Author information
Authors and Affiliations
Additional information
Partially supported by CNPq-Brazil.
About this article
Cite this article
Movasati, H. On the topology of foliations with a first integral. Bol. Soc. Bras. Mat 31, 305–336 (2000). https://doi.org/10.1007/BF01241632
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01241632