Abstract
Given a meromorphic function \(s: {\mathbb {C}}\rightarrow \mathbb {P}^{1}\), we obtain a family of fiber-preserving dominating holomorphic maps from \(\mathbb {C}^{2}\) onto \(\mathbb {C}^{2}\setminus \text{ graph }(s)\) defined in terms of the flows of complete vector fields of type \(\mathbb {C}^{*}\) and of an entire function \(h:\mathbb {C}\rightarrow \mathbb {C}\) whose graph does not meet \(\text{ graph }(s)\), which was determined by Buzzard and Lu. In particular, we prove that the dominating map constructed by these authors to prove the dominability of \(\mathbb {C}^{2}\setminus \text{ graph }(s)\) is in the above family. We also study the complement of a double section in \(\mathbb {C}\times \mathbb {P}^{1}\) in terms of a complex flow. Moreover, when \(s\) has at most one pole, we prove that there are infinitely many complete vector fields tangent to \(\text{ graph }(s)\), describing explicit families of them with all their trajectories proper and of the same type (\(\mathbb {C}\) or \(\mathbb {C}^{*}\)), if \(\text{ graph }(s)\) does not contain zeros; and families with almost all trajectories non-proper and of type \(\mathbb {C}\), or of type \(\mathbb {C}^{*}\), if \(\text{ graph }(s)\) contains zeros. We also study the dominability of \(\mathbb {C}^{2}\setminus A\) when \(A\subset \mathbb {C}^{2}\) is invariant by the flow of a complete holomorphic vector field.
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References
Brunella, M.: Sur les courbes intégrales propres des champs de vecteurs polynomiaux. Topology 37(6), 1229–1246 (1998)
Brunella, M.: Birational geometry of foliations. First Latin American Congress of Mathematicians, IMPA (2000)
Brunella, M.: Complete vector fields on the complex plane. Topology 43(2), 433–445 (2004)
Bustinduy, A.: On the entire solutions of a polynomial vector field on \(\mathbb{C}^2\). Indiana Univ. Math. J. 53, 647–666 (2004)
Buzzard, G., Lu, S.: Algebraic surfaces holomorphically dominable by \(\mathbb{C}^2\). Invent. Math. 139, 617–659 (2000)
Buzzard, G., Lu, S.: Double sections, dominating maps, and the Jacobian fibration. Am. J. Math. 122, 1061–1084 (2000)
Loray, F.: Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Prépublication IRMAR (2005) http://hal.archives-ouvertes.fr/ccsd-00016434
Suzuki, M.: Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace \(\mathbb{C}^{2}\). J. Math. Soc. Japan 26, 241–257 (1974)
Suzuki, M.: Sur les opérations holomorphes du groupe additif complexe sur l’espace de deux variables complexes. Ann. Sci. École Norm. Sup. 10, 517–546 (1977)
Suzuki, M.: Sur les opérations holomorphes de \(\mathbb{C}\) et de \(\mathbb{C}^{\ast }\) sur un espace de Stein. Lect. Notes Math. 394, 80–88 (1978)
Zaĭdenberg, M.G., Lin, VYa.: An irreducible, simply connected algebraic curve in \(\mathbb{C}^{2}\) is equivalent to a quasihomogeneous curve. Dokl. Akad. Nauk SSSR 271, 1048–1052 (1983)
Acknowledgments
I want to thank Professor Miguel Sánchez Caja and the Universidad de Granada for their invitation, during which I could finish this article. Thanks as well to Professors Junjiro Noguchi and Takeo Ohsawa to let me know the work of Buzzard and Lu. Finally, we also want to thank the referee for his suggestions that have improved this paper a lot.
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Supported by Spanish MICINN project MTM2011-26674-C02-02.
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Bustinduy, A. Complements of graphs of meromorphic functions and complete vector fields. Math. Z. 278, 1097–1112 (2014). https://doi.org/10.1007/s00209-014-1347-x
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DOI: https://doi.org/10.1007/s00209-014-1347-x