Special foliations on \(\mathbb{CP}^2\) with a unique singular point. (English) Zbl 1487.32170
Summary: In this work, we construct, for any \(d\geq 2\), a new foliation on \(\mathbb{CP}^2\) of degree \(d\) with a unique singular point of multiplicity \(d-1\) without invariant algebraic curves that contain all its separatrices. We also prove that if \(X\) is a foliation on \(\mathbb{CP}^2\) with a unique nilpotent singular point, then \(X\) has no algebraic leaves. Finally, we characterize logarithmic foliations on \(\mathbb{CP}^2\) with a unique singular point. And we give some new examples of this kind of foliations.
MSC:
| 32S65 | Singularities of holomorphic vector fields and foliations |
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
| 14B05 | Singularities in algebraic geometry |
References:
| [1] | Alcántara, CR, Foliations on of degree \(d\) with a singular point with Milnor number \(d^2+d+1\), Rev. Mat. Complut., 31, 1, 187-199 (2018) · Zbl 1381.37056 · doi:10.1007/s13163-017-0239-0 |
| [2] | Alcántara, CR; Pantaleón-Mondragón, R., Foliations on with a unique singular point without invariant algebraic curves, Geom. Dedicata, 207, 193-200 (2020) · Zbl 1445.37034 · doi:10.1007/s10711-019-00492-8 |
| [3] | Brunella, M.: Birational Geometry of Foliations. IMPA Monographs 1. Springer, ISBN: 978-3-319-14309-5; 978-3-319-14310-1 · Zbl 1073.14022 |
| [4] | Camacho, C.; Lins Neto, A.; Sad, P., Topological invariants and equidesingularization for holomorphic vector fields, J. Differ. Geom., 20, 1, 143-174 (1984) · Zbl 0576.32020 · doi:10.4310/jdg/1214438995 |
| [5] | Camacho, C.; Lins Neto, A.; Sad, P., Minimal sets of foliations on complex projective spaces, Inst. Hautes Etudes Sci. Publ. Math., 68, 1988, 187-203 (1989) · Zbl 0682.57012 |
| [6] | Cerveau, D., Formes Logarithmiques et Feuilletages non Dicritiques, J. Singul., 9, 50-55 (2014) · Zbl 1296.32010 |
| [7] | Cerveau, D.; Déserti, J.; Garba Belko, D.; Meziani, R., Géométrie classique de certains feuilletages de degré deux, Bull. Braz. Math. Soc. New Ser., 41, 2, 161-198 (2010) · Zbl 1217.37048 · doi:10.1007/s00574-010-0008-x |
| [8] | Fernández-Pérez, A.; Mol, R., Residue-type indices and holomorphic foliations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19, 3, 1111-1134 (2019) · Zbl 1436.32099 |
| [9] | Gómez-Mont, X., An algebraic formula for the index of a vector field on a hypersurface with an isolated Singularity, J. Algebr. Geom., 7, 4, 731-752 (1998) · Zbl 0956.32029 |
| [10] | Gómez-Mont, X., Ortiz-Bobadilla, L.: Sistemas dinámicos holomorfos en superficies. Aportaciones Matemáticas: Notas de Investigación [Mathematical Contributions: Research Notes], 3. Sociedad Matemática Mexicana, México (1989) · Zbl 0855.58049 |
| [11] | Joanoulou, JP, Equations de Pfaff algébriques (French) [Algebraic Pfaffian Equations] Lecture Notes in Mathematics (1979), Berlin: Springer, Berlin · Zbl 0477.58002 |
| [12] | Lins Neto, A.; Soares, MG, Algebraic Solutions of one-dimensional Foliations, J. Differ. Geom., 43, 652-673 (1996) · Zbl 0873.32031 · doi:10.4310/jdg/1214458327 |
| [13] | Meziani, R., Classification analytique d’équations différentielles \(ydy+...=0\) et espace de modules, Bol. Soc. Brasil. Mat. (N.S.), 27, 1, 23-53 (1996) · Zbl 0880.34002 · doi:10.1007/BF01246703 |
| [14] | Ploski, A., A bound for the Milnor number of plane curve singularities, Cent. Eur. J. Math., 12, 5, 688-693 (2014) · Zbl 1310.14010 |
| [15] | Stròzyna, E., The analytic and formal normal form for the nilpotent singularity. The case of generalized saddle-node, Bull. Sci. Math., 126, 7, 555-579 (2002) · Zbl 1027.34099 · doi:10.1016/S0007-4497(02)01127-2 |
| [16] | Wall, CTC, Singular Points of Plane Curves. London Mathematical Society Student Texts (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1057.14001 · doi:10.1017/CBO9780511617560 |
| [17] | Zoladek, H., New examples of holomorphic foliations without algebraic leaves, Studia Math., 131, 2, 137-142 (1998) · Zbl 0930.34077 · doi:10.4064/sm-131-2-137-142 |
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