Stability of complex foliations transverse to fibrations
HTML articles powered by AMS MathViewer
- by Fabio Santos and Bruno Scardua PDF
- Proc. Amer. Math. Soc. 140 (2012), 3083-3090 Request permission
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 $H^1(B, \operatorname {GL}(k,\mathbb R))=0$.References
- 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.
- 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, DOI 10.1007/978-1-4612-5292-4
- 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
- Rémi Langevin and Harold Rosenberg, On stability of compact leaves and fibrations, Topology 16 (1977), no. 1, 107–111. MR 461523, DOI 10.1016/0040-9383(77)90034-9
- 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, DOI 10.1090/mmono/193
- Bruno Scárdua, On complex codimension-one foliations transverse fibrations, J. Dyn. Control Syst. 11 (2005), no. 4, 575–603. MR 2170665, DOI 10.1007/s10883-005-8819-6
- B. Azevedo Scárdua, Holomorphic foliations transverse to fibrations on hyperbolic manifolds, Complex Variables Theory Appl. 46 (2001), no. 3, 219–240. MR 1869737, DOI 10.1080/17476930108815410
- Schur, I.: Über Gruppen periodischer substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1911), 619–627.
- William P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347–352. MR 356087, DOI 10.1016/0040-9383(74)90025-1
Additional Information
- Fabio Santos
- Affiliation: Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, Niteroi, Rio de Janeiro 24.020-140, Brazil
- Email: fabio@mat.uff.br
- Bruno Scardua
- Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68530, Rio de Janeiro, RJ, 21945-970, Brazil
- Email: scardua@im.ufrj.br
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3083-3090
- MSC (2010): Primary 37F75, 57R30; Secondary 57R32, 32M05
- DOI: https://doi.org/10.1090/S0002-9939-2011-11136-5
- MathSciNet review: 2917081