Proceedings of the Japan Academy, Series A, Mathematical Sciences

Semicomplete vector fields on non-Kähler surfaces

Adolfo Guillot

Full-text: Open access

Abstract

We investigate semicomplete meromorphic vector fields on complex surfaces, those where the solutions of the associated ordinary differential equations have no multivaluedness. We prove that if a non-Kähler compact complex surface has such a vector field, then, up to a bimeromorphic transformation, either the vector field is holomorphic, has a first integral or preserves a fibration. This extends previous results of Rebelo and the author to the non-Kähler setting.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 93, Number 8 (2017), 73-76.

Dates
First available in Project Euclid: 3 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.pja/1506996020

Digital Object Identifier
doi:10.3792/pjaa.93.73

Subjects
Primary: 34M45: Differential equations on complex manifolds
Secondary: 32S65: Singularities of holomorphic vector fields and foliations

Keywords
Semicompleteness Enoki surface foliation

Citation

Guillot, Adolfo. Semicomplete vector fields on non-Kähler surfaces. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 8, 73--76. doi:10.3792/pjaa.93.73. https://projecteuclid.org/euclid.pja/1506996020


Export citation

References

  • W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 4, Springer-Verlag, Berlin, 1984.
  • G. Dloussky, Structure des surfaces de Kato, Mém. Soc. Math. France (N.S.) No. 14 (1984), ii+120 pp.
  • G. Dloussky and F. Kohler, Classification of singular germs of mappings and deformations of compact surfaces of class VII$_{0}$, Ann. Polon. Math. 70 (1998), 49–83.
  • G. Dloussky, K. Oeljeklaus and M. Toma, Surfaces de la classe VII$_{0}$ admettant un champ de vecteurs. II, Comment. Math. Helv. 76 (2001), no. 4, 640–664.
  • I. Enoki, Surfaces of class VII$_{0}$ with curves, Tôhoku Math. J. (2) 33 (1981), no. 4, 453–492.
  • A. Guillot and J. Rebelo, Semicomplete meromorphic vector fields on complex surfaces, J. Reine Angew. Math. 667 (2012), 27–65.
  • M. Inoue, New surfaces with no meromorphic functions, in Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, 423–426, Canad. Math. Congress, Montreal, QC, 1975.
  • Ma. Kato, Compact complex manifolds containing “global” spherical shells. I, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), 45–84, Kinokuniya Book Store, Tokyo, 1978.
  • K. Kodaira, On compact complex analytic surfaces. I, Ann. of Math. (2) 71 (1960), 111–152.
  • K. Kodaira, On the structure of compact complex analytic surfaces. I. Amer. J. Math. 86 (1964), 751–798.
  • K. Kodaira, On the structure of compact complex analytic surfaces. II. Amer. J. Math. 88 (1966), 682–721.
  • F. Kohler, Feuilletages holomorphes singuliers sur les surfaces contenant une coquille sphérique globale, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 1, 161–182.
  • I. Nakamura, On surfaces of class VII$_{0}$ with curves, Invent. Math. 78 (1984), no. 3, 393–443.
  • P. Painlevé, Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 25 (1902), no. 1, 1–85.
  • J. C. Rebelo, Singularités des flots holomorphes, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 411–428.