Complex Algebraic Foliations
Authors
Description
The book aims to introduce the reader to the study of complex differential equations, considered here in their most general form as holomorphic foliations.
Its main guideline is a systematic and motivated presentation of the main concepts, examples and results concerning certain global aspects of holomorphic foliations, and it is useful for anyone who enjoys mathematics in general.
Target audience
Higher education
Name: Complex Algebraic Foliations
Author(s): Alcides Lins Neto e Bruno Scárdua
Pages: 316
Publication: IMPA, 2015
ISBN: 978-85-244-0415-3
Edition: 1
1 Fundamental notions
1.1 Introduction
1.2 Holomorphic foliations
1.3 Singular foliations of dimension 1
1.4 Singular foliations of codimension one
1.5 Holonomy
1.6 Singularities of holomorphic vector fields
1.7 Suspension of a group of holomorphic diffeomorphisms
1.8 Exercises from Chapter 1
2 Foliations of dimension one in spaces
2.1 The complex projective space
2.2 Foliations in complex projective spaces
2.3 Degree of foliation
2.4 Generic singularities of projective foliations
2.5 Foliations of codimension one in CP(n)
2.6 Exercises from Chapter 2
3 Algebraic solutions of foliations
3.1 Algebraic solutions
3.2 The index theorem
3.3 The Baum-Bott theorem in CP (2)
3.4 Foliations without algebraic solutions
3.5 Exercises from Chapter 3
4 Foliations with algebraic limit set
4.1 Limit sets of foliations
4.2 Germs of biholomorphisms on C, 0, with fixed point
4.3 Groups of local diffeomorphisms with discrete orbits
4.4 Virtual Holonomy
4.5 Foliations with analytic limit set
4.6 Construction of closed meromorphic forms
4.7 The Linearization Theorem
4.8 Generalizations
4.9 Exercises from Chapter 4
5 Ilyashenko’s Rigidity Theorem
5.1 Topological and analytical equivalences
5.2 Foliations with an invariant line
5.3 Conjugation and rigidity of holonomies
5.4 The set In
5.5 Leaf density
5.6 Proof of Ilyashenko’s theorem
5.7 Generalizations
5.8 Exercises from Chapter 5
6 Transverse structures of foliations
6.1 Transverse structures of foliations
6.2 Transversely affine foliations
6.3 Extended affine structures
6.4 Classification of transversely affine foliations
6.5 Soluble holonomy groups and transversely affine foliations
6.6 Transversely projective foliations
6.7 Development of a transversally projective foliation
6.8 Projective meromorphic suits
6.9 Dual foliation to a transversely projective one
6.10 Classification of transversely projective foliations
6.11 Irreducible components of spaces of foliations
6.12 Exercises from Chapter 6
7 APPENDIX – Extension theorems
7.1 Open holomorphic functions of Cn
7.2 Hartogs’ theorem
7.3 Levi’s extension theorem
7.4 The global extension theorem
Bibliographical References
Index