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Stability of complex foliations transverse to fibrations

Authors: Fabio Santos and Bruno Scardua
Journal: Proc. Amer. Math. Soc. 140 (2012), 3083-3090
MSC (2010): Primary 37F75, 57R30; Secondary 57R32, 32M05
Published electronically: December 30, 2011
MathSciNet review: 2917081
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Abstract: We prove that a holomorphic foliation of codimension $ k$ which is transverse to the fibers of a fibration and has a compact leaf with finite holonomy group is a Seifert fibration, i.e., has all leaves compact with finite holonomy. This is the case for $ C^1$-small deformations of a foliation where the original foliation exhibits a compact leaf and the base $ B$ of the fibration satisfies $ H^1(B,\mathbb{R})=0$ and $ H^1(B, \operatorname {GL}(k,\mathbb{R}))=0$.

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  • 1. Burnside, W.: On criteria for the finiteness of the order of a group of linear substitutions, Proc. London Math. Soc. (2) 3 (1905), 435-440.
  • 2. César Camacho and Alcides Lins Neto, Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman. MR 824240
  • 3. Claude Godbillon, Feuilletages, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991 (French). Études géométriques. [Geometric studies]; With a preface by G. Reeb. MR 1120547
  • 4. Rémi Langevin and Harold Rosenberg, On stability of compact leaves and fibrations, Topology 16 (1977), no. 1, 107–111. MR 0461523
  • 5. Toshio Nishino, Function theory in several complex variables, Translations of Mathematical Monographs, vol. 193, American Mathematical Society, Providence, RI, 2001. Translated from the 1996 Japanese original by Norman Levenberg and Hiroshi Yamaguchi. MR 1818167
  • 6. Bruno Scárdua, On complex codimension-one foliations transverse fibrations, J. Dyn. Control Syst. 11 (2005), no. 4, 575–603. MR 2170665, 10.1007/s10883-005-8819-6
  • 7. B. Azevedo Scárdua, Holomorphic foliations transverse to fibrations on hyperbolic manifolds, Complex Variables Theory Appl. 46 (2001), no. 3, 219–240. MR 1869737
  • 8. Schur, I.: Über Gruppen periodischer substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1911), 619-627.
  • 9. William P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347–352. MR 0356087

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Additional Information

Fabio Santos
Affiliation: Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, Niteroi, Rio de Janeiro 24.020-140, Brazil

Bruno Scardua
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68530, Rio de Janeiro, RJ, 21945-970, Brazil

Keywords: Foliation, suspension, global holonomy, Seifert fibration.
Received by editor(s): June 6, 2010
Received by editor(s) in revised form: March 19, 2011
Published electronically: December 30, 2011
Communicated by: Brooke Shipley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.