Holomorphic Foliations with Singularities: Key Concepts and Modern Results (Latin American Mathematics Series)

This concise textbook gathers together key concepts and modern results on the theory of holomorphic foliations with singularities, offering a compelling vision on how the notion of foliation, usually linked to real functions and manifolds, can have an important role in the holomorphic world, as shown by modern results from mathematicians as H. Cartan, K. Oka, T. Nishino, and M. Suzuki. The text starts with a gentle presentation of the classical notion of foliations, advancing to holomorphic foliations and then holomorphic foliations with singularities. The theory behind reduction of singularities is described in detail, as well the cases for dynamics of a local diffeomorphism and foliations on complex projective spaces. A final chapter brings recent questions in the field, as holomorphic flows on Stein spaces and transversely homogeneous holomorphic foliations, along with a list of open questions for further study and research. Selected exercises at the end of each chapter help the reader to grasp the theory. Graduate students in Mathematics with a special interest in the theory of foliations will especially benefit from this book, which can be used as supplementary reading in Singularity Theory courses, and as a resource for independent study on this vibrant field of research. 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