Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Construction of vector fields and Riccati foliations associated to groups of projective automorphisms

Authors: Fabio Santos and Bruno Scárdua
Journal: Conform. Geom. Dyn. 14 (2010), 154-166
MSC (2010): Primary 37F75, 32S65; Secondary 32M25, 32M05
Published electronically: June 2, 2010
MathSciNet review: 2652067
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Our main result states that given a finitely generated subgroup $ G$ of $ \operatorname{Aut}(\mathbb{C} P (2))$, there is an algebraic foliation $ \mathcal{F}$ on a complex projective $ 3$-manifold $ M^3$ with a bundle structure over $ \mathbb{C} P(1)$ and fiber $ \mathbb{C} P(2)$, such that $ \mathcal{F}$ is transverse to almost every fiber of the bundle and with global holonomy conjugate to $ G$.

References [Enhancements On Off] (What's this?)

  • 1. Camacho, César; Lins Neto, Alcides, Geometry theory of foliations. Translated from the Portuguese by Sue E. Goodman. Birkhäuser Boston, Inc., Boston, MA, 1985. vi + 205 pp. MR 824240 (87a:57029)
  • 2. Godbillon, Claude, Feuilletages. Études géométriques. With a preface by G. Reeb. Progress in Mathematics, 98. Birkhäuser Verlag, Basel, 1991. MR 1120547 (93i:57038)
  • 3. Griffiths, Phillip; Harris, Joseph, Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. xiv + 813 pp. MR 1288523 (95d:14001)
  • 4. Gunning, Robert C., Introduction to holomorphic functions of several variables. Vol. I. Function theory. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. xx + 203 pp. MR 1052649 (92b:32001a)
  • 5. Lins Neto, Alcides, Construction of singular holomorphic vector fields and foliations in dimension two. J. Differential Geom. 26 (1987), no. 1, 1 - 31. MR 892029 (88f:32047)
  • 6. Pan, I.; Sebastiani, M., Les Équations Différentielles Algébriques et les Singularités Mobiles. Monografias de Matemática do IMPA. Rio de Janeiro, Brazil, 2005. MR 2101092 (2006f:32042b)
  • 7. Pan, I.; Sebastiani, M., Sur les équations différentielles algébriques admettant des solutions avec une singularité essentielle. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1621-1633. MR 1871283 (2002m:34132)
  • 8. Scárdua, Bruno, On complex codimension-one foliations transverse fibrations. J. Dyn. Control Syst. 11 (2005), no. 4, 575-603. MR 2170665 (2006g:32049)
  • 9. Scárdua, Bruno, Holomorphic foliations transverse to fibrations on hyperbolic manifolds. Complex Variables Theory Appl. 46 (2001), no. 3, 219-240. MR 1869737 (2002m:32048)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 37F75, 32S65, 32M25, 32M05

Retrieve articles in all journals with MSC (2010): 37F75, 32S65, 32M25, 32M05

Additional Information

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

Bruno Scárdua
Affiliation: Instituto de Matematica, Universidade Federal do Rio de Janeiro, CP. 68530-Rio de Janeiro-RJ, 21945-970, Brazil

Keywords: Holomorphic foliation, holonomy, projective automorphism
Received by editor(s): August 27, 2009
Published electronically: June 2, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society