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 relating to certain global aspects of holomorphic foliations, making it useful to all who appreciate mathematics in general.
Target audience
Higher education
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 concepts
1.1 Introduction
1.2 Holomorphic Foliations
1.3 Single foliations of dimension 1
1.4 Single 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 Chapter 1 Exercises
2. One-dimensional foliations 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 Chapter 2 Exercises
3 Algebraic solutions of foliations
3.1 Algebraic solutions
3.2 The Index Theorem
3.3 Baum-Bott's Theorem in CP (2)
3.4 Foliations without algebraic solutions
3.5 Chapter 3 Exercises
4. Foliations with algebraic limit set
4.1 Foliation boundary sets
4.2 Germs of biholomorphisms in C , 0, with a fixed point
4.3 Groups of local diffeomorphisms with discrete orbits
4.4 Virtual Holonomy
4.5 Foliations with analytical limit set
4.6 Construction of closed meromorphic forms
4.7 The Linearization Theorem
4.8 Generalizations
4.9 Chapter 4 Exercises
5. Ilyashenko's Stiffness Theorem
5.1 Topological and Analytic Equivalences
5.2 Foliations with an invariant line
5.3 Conjugation and rigidity of holonomies
5.4 The set I n
5.5 Leaf Density
5.6 Proof of Ilyashenko's Theorem
5.7 Generalizations
5.8 Chapter 5 Exercises
6. Cross-sectional structures of foliations
6.1 Cross-sectional structures of foliations
6.2 Transversely related foliations
6.3 Extended related structures
6.4 Classification of transversely related foliations
6.5 Soluble holonomy groups and transversely related foliations
6.6 Transversely projecting foliations
6.7 Development of a transversely projective foliation
6.8 Projective meromorphic suits
6.9 Dual foliation to a transversely projective
6.10 Classification of transversely projecting foliations
6.11 Irreducible components of foliation spaces
6.12 Chapter 6 Exercises
7 APPENDIX – Extension Theorems
7.1 Holomorphic functions on open sets of C n
7.2 Hartog's Theorem
7.3 Levi's Extension Theorem
7.4 The global extension theorem
Bibliographic References
Index