Holomorphic Foliations with Singularities: Key Concepts and Modern Results

Preface
Contents
1 The Classical Notions of Foliations
	1.1 Definition of Foliation
	1.2 Other Definitions of Foliation
	1.3 Frobenius Theorem
	1.4 Holonomy
	1.5 Exercises
2 Some Results from Several Complex Variables
	2.1 Some Extension Theorems from Several Complex Variables
	2.2 Levi\'s Global Extension Theorem
	2.3 Exercises
3 Holomorphic Foliations: Non-singular Case
	3.1 Basic Concepts
	3.2 Examples
	3.3 The Identity Principle for Holomorphic Foliations
	3.4 Exercises
4 Holomorphic Foliations with Singularities
	4.1 Linear Vector Fields on the Plane
	4.2 One-Dimensional Foliations with Isolated Singularities
	4.3 Differential Forms and Vector Fields
	4.4 Codimension One Foliations with Singularities
	4.5 Analytic Leaves
	4.6 Two Extension Lemmas for Holomorphic Foliations
	4.7 Kupka Singularities and Simple Singularities
	4.8 Exercises
5 Holomorphic Foliations Given by Closed 1-Forms
	5.1 Foliations Given by Closed Holomorphic 1-Forms
		5.1.1 Holonomy of Foliations Defined by Closed Holomorphic 1-Forms
	5.2 Foliations Given by Closed Meromorphic 1-Forms
		5.2.1 Holonomy of Foliations Defined by Closed meromorphic 1-Forms
	5.3 Exercises
6 Reduction of Singularities
	6.1 Irreducible Singularities
	6.2 Poincaré and Poincaré–Dulac Normal Forms
	6.3 Blow-up at the Origin (Quadratic Blow-up)
	6.4 Blow-up on Surfaces
		6.4.1 Resolution of Curves
	6.5 Blow-up of a Singular Point of a Foliation
	6.6 Irreducible Singularities
	6.7 Separatrices: Dicriticalness and Existence
	6.8 Holonomy and Analytic Classification
		6.8.1 Holonomy of Irreducible Singularities
		6.8.2 Holonomy and Analytic Classification of Irreducible Singularities
	6.9 Examples
	6.10 Exercises
7 Holomorphic First Integrals
	7.1 Mattei–Moussu Theorem
	7.2 Groups of Germs of Holomorphic Diffeomorphisms
	7.3 Irreducible Singularities
	7.4 The Case of a Single Blow-up
	7.5 The General Case
	7.6 Exercises
8 Dynamics of a Local Diffeomorphism
	8.1 Hyperbolic Case
	8.2 Parabolic Case
	8.3 Elliptic Case
	8.4 Exercises
9 Foliations on Complex Projective Spaces
	9.1 The Complex Projective Plane and Foliations
	9.2 The Theorem of Darboux–Jouanolou
	9.3 Foliations Given by Closed 1-Forms
	9.4 Riccati Foliations
	9.5 Examples of Foliations on C P(2)
	9.6 Example of an Action of a Low-dimensional Lie Algebra
	9.7 A Family of Foliations on C P(3) Not Coming from Plane Foliations
	9.8 Exercises
10 Foliations with Algebraic Limit Sets
	10.1 Limit Sets of Foliations
	10.2 Groups of Germs of Diffeomorphisms with Finite Limit Set
	10.3 Virtual Holonomy Groups
	10.4 Construction of Closed Meromorphic Forms
	10.5 The Linearization Theorem
	10.6 Examples
	10.7 Exercises
11 Some Modern Questions
	11.1 Holomorphic Flows on Stein Spaces
		11.1.1 Suzuki\'s Theory
		11.1.2 Proof of the Global Linearization Theorem
	11.2 Real Transverse Sections of Holomorphic Foliations
	11.3 Non-trivial Minimal Sets of Holomorphic Foliations
	11.4 Transversely Homogeneous Holomorphic Foliations
		11.4.1 Transversely Lie Foliations
	11.5 Transversely Affine Foliations
	11.6 Transversely Projective Foliations
		11.6.1 Development of a Transversely Projective Foliation—Touzet\'s Work
		11.6.2 Projective Structures and Differential Forms
			Proof of Proposition 11.6.5
			Classification of Projective Foliations: Moderate Growth on Projective Manifolds
12 Miscellaneous Exercises and Some Open Questions
	12.1 Miscellaneous Exercises
	12.2 Some Open Questions
Bibliography
Index