Holomorphic foliations with singularities : key concepts and modern results / / Bruno Scárdua

• Autore: Scárdua Bruno
• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (172 pages)
• Disciplina: 514.72
• Soggetto topico: Foliations (Mathematics); Domains of holomorphy; Algebraic topology; Foliacions (Matemàtica)
• Soggetto genere / forma: Llibres electrònics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Holomorphic Foliations with Singularities
• ISBN: 3-030-76705-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466563603316
• Lo trovi qui: Univ. di Salerno
• Serie: Latin American Mathematics