On foliations with rational first integral
DOI:
https://doi.org/10.4067/S0716-09172005000200006Keywords:
Compact complex surfaces, Holomorphic foliation, Invariant compact curves.Abstract
We use the Kodaira’s classification of relatively minimal elliptic fi- brations to prove that a holomorphic foliation in CP2 with a dicritical compact elliptic curve has rational first integral.
References
[1] P. Baum and R. Bott, On the zeroes of meromorphic foliations, Essais en l’honeur de G. De Rham, pp. 29-47, (1970).
[2] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Springer Ergebnisse 4, (1984).
[3] M. Brunella,Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. E. N. S. 4, Vol. 30, pp. 569-594, (1997).
[4] C. Camacho, A. Lins Neto and P. Sad, Topological invariants and equidesingularization for holomorphic vector field, Jour. Diff. Geom. vol. 20, pp. 143—174, (1984).
[5] D. Cerveau and A. Lins Neto, Holomorphic foliation in CP2 having an invariant algebraic curve, Ann. Inst. Fourier, Grenoble 41, 4, pp. 883—903, (1991).
[6] A. Lins Neto Some examples for the Poincaré and Painlevé problems, Ann. Sci. Ecole Norm. Sup. (4) 35, No. 2, pp. 231—266, (2002).
[7] S. Licanic, An upper bound for the total sum of the Baum-Bott indexes of a holomorphic foliation and the Poincare Problem, Hokkaido Mathematical Journal Vol. 33, No.3, October, (2004).
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[10] H. Poincaré Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré , Rend. Circ. Mat. Palermo 11, pp. 193—239, (1897).
[2] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Springer Ergebnisse 4, (1984).
[3] M. Brunella,Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. E. N. S. 4, Vol. 30, pp. 569-594, (1997).
[4] C. Camacho, A. Lins Neto and P. Sad, Topological invariants and equidesingularization for holomorphic vector field, Jour. Diff. Geom. vol. 20, pp. 143—174, (1984).
[5] D. Cerveau and A. Lins Neto, Holomorphic foliation in CP2 having an invariant algebraic curve, Ann. Inst. Fourier, Grenoble 41, 4, pp. 883—903, (1991).
[6] A. Lins Neto Some examples for the Poincaré and Painlevé problems, Ann. Sci. Ecole Norm. Sup. (4) 35, No. 2, pp. 231—266, (2002).
[7] S. Licanic, An upper bound for the total sum of the Baum-Bott indexes of a holomorphic foliation and the Poincare Problem, Hokkaido Mathematical Journal Vol. 33, No.3, October, (2004).
[8] H. Poincaré Sur l´intégration algébrique des équations différentielles, C. R. Acad. Sci. 112, pp. 761—764, (1891).
[9] H. Poincaré Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré, Rend. Circ. Mat. Palermo 5, pp. 161—191, (1891).
[10] H. Poincaré Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré , Rend. Circ. Mat. Palermo 11, pp. 193—239, (1897).
Published
2017-04-20
How to Cite
[1]
S. Licanic, “On foliations with rational first integral”, Proyecciones (Antofagasta, On line), vol. 24, no. 2, pp. 175-179, Apr. 2017.
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