Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T13:40:11.304Z Has data issue: false hasContentIssue false
  • English
  • Français

ONE-DIMENSIONAL LIE FOLIATIONS WITH GENERIC SINGULARITIES IN COMPLEX DIMENSION THREE

Part of: Differential topology

Published online by Cambridge University Press:  28 September 2011

ALBETÃ MAFRA  and
BRUNO SCARDUA
ALBETÃ MAFRA
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970-Rio de Janeiro, Brazil (email: [email protected])
BRUNO SCARDUA*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970-Rio de Janeiro, Brazil (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that a germ of a one-dimensional holomorphic foliation with a generic singularity in dimension two or three that exhibits a Lie group transverse structure in the complement of some codimension one analytic subset is logarithmic, that is, given by a system of closed meromorphic one-forms with simple poles. In the global context, we prove that a foliation by curves in a three-dimensional complex manifold with generic singularities and a Lie group transverse structure off a codimension one analytic subset is logarithmic; that is, it is given by a system of closed meromorphic one-forms with simple poles.

Keywords

holomorphic foliation with singularitiesLie transverse structure

MSC classification

Secondary: 57R30: Foliations; geometric theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 1 , August 2011 , pp. 13 - 28
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Behague, A. and Scárdua, B., ‘Foliations invariant under Lie group transverse actions’, Monatsh. Math. 153 (2008), 295–308.CrossRefGoogle Scholar
[2]Blumenthal, R., ‘Transversely homogeneous foliations’, Ann. Inst. Fourier (Grenoble) 29 (1979), 143–158.CrossRefGoogle Scholar
[3]Brjuno, A. D., ‘Analytical form of differential equations’, Trans. Moscow Math. Soc. 25 (1971), 131–288.Google Scholar
[4]Calvo Andrade, O., ‘Irreducible components of the space of holomorphic foliations’, Math. Ann. 299 (1994), 751–767.CrossRefGoogle Scholar
[5]Camacho, C. and Lins Neto, A., Geometric Theory of Foliations (Birkhäuser, Berlin, 1985).CrossRefGoogle Scholar
[6]Camacho, C. and Sad, P., ‘Invariant varieties through singularities of holomorphic vector fields’, Ann. of Math. (2) 115 (1982), 579–595.CrossRefGoogle Scholar
[7]Camacho, C. and Scárdua, B., ‘Holomorphic foliations with Liouvillian first integrals’, Ergod. Th. & Dynam. Sys. 21 (2001), 717–756.CrossRefGoogle Scholar
[8]Dulac, H., ‘Solutions d’un système d’équations différentielles dans le voisinage de valeurs singulières’, Bull. Soc. Math. France 40 (1912), 324–383.CrossRefGoogle Scholar
[9]Godbillon, C., Feuilletages. Etudes Géométriques, Progress in Mathematics, 98 (Birkhäuser, Basel, 1991).Google Scholar
[10]Gunning, R. C., Introduction to Holomorphic Functions of Several Variables. Vol. I. Function Theory, The Wadsworth & Brooks/Cole Mathematics Series (Wadsworth & Brooks/Cole, Pacific Grove, CA, 1990).Google Scholar
[11]Gunning, R. C., Introduction to Holomorphic Functions of Several Variables. Vol. II. Local Theory, The Wadsworth & Brooks/Cole Mathematics Series (Wadsworth & Brooks/Cole, Pacific Grove, CA, 1990).Google Scholar
[12]Gunning, R. C. and Rossi, H., Analytic Functions of Several Complex Variables (Prentice Hall, Englewood Cliffs, NJ, 1965).Google Scholar
[13]Martinet, J. and Ramis, J.-P., ‘Problème de modules pour des équations différentielles non linéaires du premier ordre’, Publ. Math. Inst. Hautes Études Sci. 55 (1982), 63–124.CrossRefGoogle Scholar
[14]Martinet, J. and Ramis, J.-P., ‘Classification analytique des équations différentielles non linéaires résonnants du premier ordre’, Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 571–621.CrossRefGoogle Scholar
[15]Mattei, J-.F. and Moussu, R., ‘Holonomie et intégrales premières’, Ann. Sci. Éc. Norm. Supér. (4) 13 (1980), 469–523.CrossRefGoogle Scholar
[16]Scárdua, B., ‘Transversely affine and transversely projective holomorphic foliations’, Ann. Sci. Éc. Norm. Supér. 30 (1997), 169–204.CrossRefGoogle Scholar
[17]Scárdua, B., ‘Integration of complex differential equations’, J. Dyn. Control Syst. 5 (1999), 1–50.CrossRefGoogle Scholar
[18]Seidenberg, A., ‘Reduction of singularities of the differential equation A dy=B dx’, Amer. J. Math. 90 (1968), 248–269.CrossRefGoogle Scholar
[19]Siegel, C. L., ‘Iteration of analytic functions’, Ann. of Math. (2) 43 (1942), 607–612.CrossRefGoogle Scholar
[20]Siu, Y., Techniques of Extension of Analytic Objects (Marcel Dekker, New York, 1974).Google Scholar
[21]Touzet, F., ‘Sur les feuilletages holomorphes transversalement projectifs’, Ann. Inst. Fourier (Grenoble) 53 (2003), 815–846.CrossRefGoogle Scholar