Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T15:41:32.250Z Has data issue: false hasContentIssue false
  • English
  • Français

HOLOMORPHIC FLOWS WITH PERIODIC ORBITS ON STEIN SURFACES

Part of: Automorphic functions Complex spaces with a group of automorphisms Holomorphic convexity Complex dynamical systems

Published online by Cambridge University Press:  22 June 2010

BRUNO SCÁRDUA
BRUNO SCÁRDUA*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro - RJ, Caixa Postal 68530, 21.945-970 Rio de Janeiro-RJ, Brazil (email: scardua@im.ufrj.br)
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.

In this paper we study the classification of holomorphic flows on Stein spaces of dimension two. We assume that the flow has periodic orbits, not necessarily with a same period. Then we prove a linearization result for the flow, under some natural conditions on the surface.

Keywords

Stein manifoldholomorphic flowdicritical singularityholomorphic foliation

MSC classification

Secondary: 32E10: Stein spaces, Stein manifolds 37F75: Holomorphic foliations and vector fields 32M25: Complex vector fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 177 - 186
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Brunella, M., ‘Some remarks on parabolic foliations’, in: Geometry and Dynamics, Contemporary Mathematics, 389 (American Mathematical Society, Providence, RI, 2005), pp. 91–102.CrossRefGoogle Scholar
[2]Camacho, C., Movasati, H. and Scárdua, B., ‘-actions on Stein analytic spaces with isolated singularities’, J. Geom. Anal. 19(2) (2009), 244–260.CrossRefGoogle Scholar
[3]Camacho, C. and Sad, P., ‘Invariant varieties through singularities of holomorphic vector fields’, Ann. of Math. (2) 115 (1982), 579–595.CrossRefGoogle Scholar
[4]Camacho, C. and Scárdua, B., ‘Dicritical holomorphic flows on Stein manifolds’, Arch. Math. 89 (2007), 339–349.CrossRefGoogle Scholar
[5]Camacho, C. and Scárdua, B., ‘Nondicritical ℂ*-actions on two-dimensional Stein manifolds’, Manuscripta Math. 129(1) (2009), 91–98.CrossRefGoogle Scholar
[6]Camacho, C. and Scárdua, B., ‘Actions of the groups ℂ and ℂ* on Stein varieties.’, Geom. Dedicata 139 (2009), 5–14.CrossRefGoogle Scholar
[7]Cerveau, D. and Scárdua, B., ‘Complete polynomial vector fields in two complex variables’, Forum Math. 17(3) (2005), 407–430.CrossRefGoogle Scholar
[8]Hausen, J., ‘Zur Klassifikation glatter kompakter ℂ*-Flächen’, Math. Ann. 301 (1995).CrossRefGoogle Scholar
[9]Hausen, J., ‘Holomorphe ℂ*-Operationen auf komplexen Flächen’, Dissertation, Konstanzer Schriften in Mathematik und Informatik 11, 1996. Available at http://www.inf.uni-konstanz.de/Preprints/papers/1996/preprint-011.ps.Google Scholar
[10]Martinet, J. and Ramis, J.-P., ‘Problème de modules pour des équations différentielles non lineaires du premier ordre’, Publ. Math. Inst. Hautes Études Sci. 55 (1982), 63–124.CrossRefGoogle Scholar
[11]Martinet, J. and Ramis, J.-P., ‘Classification analytique des équations différentielles non lineaires resonnants du premier ordre’, Ann. Sci. École Norm. Sup. 16 (1983), 571–621.CrossRefGoogle Scholar
[12]Mattei, J. F. and Moussu, R., ‘Holonomie et intégrales premières’, Ann. Sci. École Norm. Sup. 13 (1980), 469–523.CrossRefGoogle Scholar
[13]Nishino, T., ‘Nouvelles recherches sur les fonctions entires de plusieurs variables complexes. I’, J. Math. Kyoto Univ. 8 (1968), 49–100.Google Scholar
[14]Scárdua, B., ‘On the classification of holomorphic flows and Stein surfaces’, Complex Var. Elliptic Equ. 52(1) (2007), 79–83.CrossRefGoogle Scholar
[15]Seade, J., On the Topology of Isolated Singularities in Analytic Spaces, Progress in Mathematics, 241 (Birkhäuser, Basel, 2006).Google Scholar
[16]Seidenberg, A., ‘Reduction of singularities of the differential equation A dy=B dx’, Amer. J. Math. 90 (1968), 248–269.CrossRefGoogle Scholar
[17]Snow, D., ‘Reductive group actions on Stein spaces’, Math. Ann 259 (1982), 79–97.CrossRefGoogle Scholar
[18]Suzuki, M., ‘Sur les opérations holomorphes de ℂ et de ℂ* sur un espace de Stein’, in: Séminaire Norguet, Lecture Notes, 670 (Springer, Berlin, 1977), pp. 80–88.Google Scholar
[19]Suzuki, M., ‘Sur les opérations holomorphes du groupe additif complexe sur l’espace de deux variables complexes’, Ann. Sci. École Norm. Sup. (4) 10 (1977), 517–546.CrossRefGoogle Scholar
[20]Yamaguchi, H., ‘Parabolicité d’une fonction entière’, J. Math. Kyoto Univ. 16(1) (1976), 71–92.Google Scholar
[21]Yamaguchi, H., ‘Calcul des variations analytiques’, Japan. J. Math. (N.S.) 7(2) (1981), 319–377.CrossRefGoogle Scholar