Handbook of Geometry and Topology of Singularities V

• Record Nr.: UNINA9910865271703321
• Autore: Cano Felipe
• Pubbl/distr/stampa: Cham : , : Springer International Publishing AG, , 2024; ©2024
• ISBN: 3-031-52481-0
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (531 pages)
• Altri autori (Persone): Cisneros-MolinaJosé Luis; Dũng TrángLê; SeadeJosé
• Disciplina: 516.35
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Foreword: Theory of Complex Foliations in Moscow and Outside -- Preface -- Contents -- Contributors -- 1 Holomorphic Foliations: Singularities and Local Geometric Aspects -- 1.1 Introduction -- 1.2 The Notion of Holomorphic Foliation -- 1.2.1 Motivation -- 1.2.2 Definition of Holomorphic Foliation -- 1.2.3 Other Definitions of Foliation -- 1.2.4 Frobenius Theorem -- 1.2.5 Examples of Holomorphic Foliations -- 1.2.6 Holonomy -- 1.2.7 The Identity Principle for Holomorphic Foliations -- 1.3 Holomorphic Foliations with Singularities -- 1.3.1 Linear Vector Fields on the Plane -- 1.3.2 One-Dimensional Foliations with Isolated Singularities -- 1.3.3 Differential Forms and Vector Fields -- 1.3.4 Codimension One Foliations with Singularities -- 1.3.5 Analytic Leaves -- 1.3.6 Two Extension Lemmas for Holomorphic Foliations -- 1.3.7 Kupka Singularities and Simple Singularities -- 1.4 Reduction of Singularities: The Blow-Up Method -- 1.4.1 Germs of Singularities in Dimension Two -- 1.4.2 Nondegenerate Singularities -- 1.4.3 The Blow-Up Method and Resolution of Curves -- 1.4.4 Separatrices: Dicricity and Existence -- 1.4.5 Seidenberg's Theorem -- 1.4.6 Irreducible Singularities -- 1.4.7 Holonomy and Analytic Classification -- 1.5 Holomorphic First Integrals: Theorem of Mattei-Moussu -- 1.5.1 Mattei-Moussu Theorem -- 1.5.2 Groups of Germs of Holomorphic Diffeomorphisms -- 1.5.3 Irreducible ; Singularities -- 1.5.4 The Case of a Single Blow-Up -- 1.5.5 The General Case -- 1.6 Holomorphic Foliations Given by Closed 1-Forms -- 1.6.1 Foliations Given by Closed Holomorphic 1-Forms -- 1.6.2 Foliations Given by Closed Meromorphic 1-Forms -- 1.6.3 Holonomy of Foliations Defined by Closed Meromorphic 1-Forms -- 1.6.4 The Integration Lemma -- 1.7 Linearization of Foliations -- 1.7.1 Virtual Holonomy Groups -- 1.7.2 Abelian Groups and Linearization.; 1.7.3 Construction of Closed Meromorphic Forms -- 1.7.4 Proof of the Linearization Theorem -- References -- 2 Persistence, Uniformizanion and Holonomy -- 2.1 Introduction -- 2.1.1 Limit and Identical Complex Cycles: Holonomy Map -- 2.1.2 Persistence Property -- 2.1.3 Plan of the Paper -- 2.2 Simultaneous Uniformization Problem -- 2.2.1 Manifold of Universal Covers Over the Leaves -- 2.2.2 What Is Simultaneous Uniformization? -- 2.2.3 Existence and Non-Existence of Simultaneous Uniformization -- 2.3 Conditionary Persistence Theorem for Identical Cycle -- 2.3.1 A Conditionary Persistence Theorem -- 2.3.2 Persistence Domain for an Identical Cycle -- 2.3.3 Tameness on Disks -- 2.3.4 Some Auxiliary Results -- 2.3.5 Isomorphism of Canonical Skew Cylinders -- 2.3.6 Persistence of Identical Cycles -- 2.4 Destruction of Identical Cycle for Analytic Foliations of a Closed Two-Dimensional Manifold in C5 -- 2.5 Persistence of Complex Limit Cycles -- 2.5.1 Persistence Domain of Complex Limit Cycles -- 2.5.2 Statement of the Persistence Theorem -- 2.5.3 Boundary Leaves -- 2.5.4 The Induced Cylinder and Its Deck Transformation -- 2.5.5 Geometric Interpretation of the Map Fb -- 2.6 Unifomizability: Pro e Contra -- 2.6.1 Elementary Example -- 2.6.2 Main Example -- 2.6.3 Non-Uniformizable Algebraic Foliations -- 2.6.4 Uniformization Conjectures -- 2.6.5 Algebraic Families and Simultaneous Uniformization -- 2.7 Non-Extendable Holonomy Map -- 2.7.1 Limit Points of a Contracting Semigroup as Singular Points of Holonomy -- 2.7.2 A Linear Non-Homogeneous Equation -- 2.7.3 A Non-Conditional Persistence Theorem -- References -- 3 Holomorphic Foliations and Vector Fields with Degenerated Singularity in (C2,0) -- 3.1 Local Holomorphic Vector Fields and Foliations with Degenerated Singular Point in (C2, 0) -- 3.1.1 Introduction -- 3.2 Basic Results -- 3.2.1 Blow-Up of (C2,0).; 3.2.2 Blow-Up of Germs of Vector Fields in Vn+1 -- 3.3 Rigidity Theorems for Generic Non-Dicritical Foliations and Vector Fields in (C2,0) -- 3.3.1 Rigidity for Foliations of Generic Non-Dicritical Germs of Vector Fields -- 3.3.2 Rigidity for Non-Dicritical Germs of Vector Fields -- 3.4 Rigidity Theorems for Generic Dicritical Foliations and Vector Fields in (C2,0) -- 3.4.1 Rigidity for Foliations Defined by Generic Dicritical Germs of Vector Fields -- 3.4.2 Rigidity for Generic Dicritical Germs of Vector Fields -- 3.5 Formal Normal Forms for Generic Vector Fields and Foliations with Degenerate Singularity in (C2,0) -- 3.5.1 Formal Normal Forms for Generic Holomorphic Dicritical Foliations and Vector Fields in (C2,0) -- 3.5.2 Formal Normal Forms for Generic Holomorphic Non-Dicritical Foliations in (C2,0) -- 3.6 Thom's Invariants for Generic Non-Dicritical and Dicritical Foliations in (C2,0) -- 3.6.1 Thom's Invariants for Generic Holomorphic Non-Dicritical Foliations in (C2,0) -- 3.6.2 Realization Theorem: Independence of the Invariants vc and [Gv] -- 3.6.3 Thom's Invariants for Generic Holomorphic Dicritical Foliations in (C2,0) -- 3.7 Analytic Normal Forms of Germs of Foliations with Degenerated Singularity -- 3.7.1 Formal and Analytic Normal Forms of Germs of Holomorphic Non-Dicritical Foliations -- 3.7.2 Analytic Normal Forms of Germs of Generic Holomorphic Dicritical Foliations -- 3.8 Geometric Interpretation of Thom's Parametric Invariants -- References -- 4 Topology of Singular ; Foliation Germs in C2 -- 4.1 Introduction -- 4.2 Separatrices and Separators -- 4.2.1 Graph Decomposition of the Complement of a Germ of Curve -- 4.2.2 Separatrices -- 4.2.3 Separators and Dynamical Decomposition -- 4.3 Incompressibility of Leaves -- 4.3.1 Foliated Connectedness and a Foliated Van Kampen Theorem -- 4.3.2 Construction of Foliated Blocks -- 4.4 Examples.; 4.4.1 Dicritical Cuspidal Singularity -- 4.4.1.1 Fundamental Group of the Complement of S -- 4.4.1.2 Compressible Leaves -- 4.4.1.3 Appropiate Curve -- 4.4.2 Foliations Which Are Not Generalized Curves -- 4.5 Monodromy of Singular Foliations -- 4.5.1 Ends of Leaves Space of Reduced Foliations -- 4.5.1.1 Poincaré Type Singularity λ R -- 4.5.1.2 Non-Linearizable Resonant Saddles -- 4.5.1.3 Real Saddles λR&lt -- 0 -- 4.5.1.4 Non-Reduced Logarithmic Singularities -- 4.5.2 Complex Structure on Leaf Spaces -- 4.5.3 Extended Holonomy Along Geometric Blocks of the Foliation -- 4.5.4 Monodromy Representation of a Singular Foliation -- 4.5.5 Monodromy vs Holonomy Conjugacies -- 4.5.6 Classification Theorem -- 4.6 Topological Invariance of Camacho-Sad Indices -- 4.6.1 Camacho-Sad Index -- 4.6.2 Different types of Dynamical Components -- 4.6.3 Small Dynamical Components -- 4.6.4 Big Dynamical Components -- 4.6.5 Peripheral Structure and Index Invariance Theorem -- 4.7 Excellence Theorem and Topological Moduli Space -- 4.7.1 Excellence Theorem -- 4.7.2 Classification Problem: Complete Families and Moduli Space -- References -- 5 Jacobian and Polar Curves of Singular Foliations -- 5.1 Introduction -- 5.2 Generalized Curve Foliations and Logarithmic Models -- 5.2.1 Logarithmic Models -- 5.2.2 Camacho-Sad Index Relative to Singular Separatrices -- 5.3 Polar and Jacobian Intersection Multiplicities -- 5.4 Equisingularity Data of a Plane Curve -- 5.4.1 Equisingularity Data of an Irreducible Curve -- 5.4.2 Equisingularity Data of a Curve with Several Branches -- 5.4.3 Ramification -- 5.5 Topological Properties of Polar Curves of Foliations -- 5.5.1 The Case of Non-Singular Separatrices -- 5.5.2 General Case -- 5.6 Topological Properties of Jacobian Curves of Foliations -- 5.6.1 The Case of Non-Singular Separatrices -- 5.6.2 Jacobian Curve: General Case.; 5.7 Analytic Invariants of Irreducible Plane Curves -- References -- 6 Rolle Models in the Real and Complex World -- 6.1 Rolle Lemma, Virgin Flavor -- 6.1.1 First Year Calculus Revisited -- 6.1.2 Rolle Inequality and Descartes Rule of Signs -- 6.1.3 Main Building Block of Elementary Fewnomial Theory -- 6.2 Rolle Theorem and Real ODE's -- 6.2.1 De la Vallée Poussin Theorem and Higher Order Equations -- 6.2.2 Real Meandering Theorem -- 6.2.3 Maximal Tangency Order and the Gabrielov-Khovanskii Theorem -- 6.2.4 Meandering of Curves in the Euclidean Space -- 6.2.4.1 Rolle Theorem in Rn -- 6.2.5 Voorhoeve Index -- 6.2.5.1 Integral Frenet Curvatures and Spatial Meandering -- 6.2.5.2 Non-Oscillating Curves in Rn -- 6.2.6 Spatial Curves vs. Linear Ordinary Differential Equations -- 6.3 Counting Complex Roots -- 6.3.1 Kim Theorem -- 6.3.2 Jensen Inequality -- 6.3.3 Bernstein Index -- 6.3.3.1 On the Order of Quantifiers: How to Understand the Inequalities Below -- 6.3.3.2 Definition of the Bernstein Index -- 6.3.4 Variation of Argument of Solutions of Complex-Valued Linear Equations -- 6.3.5 Rolle and Triangle Inequalities for the Bernstein Index -- 6.3.5.1 Application to Pseudopolynomials -- 6.3.6 Bernstein Index for Power Series -- 6.3.7 Singular Points and Rolle Theory for Difference Operators -- 6.3.7.1 Fuchsian Singularities -- 6.3.7.2 Argument Principle for Unbounded Domains: Petrov Difference Operators and the Associated Rolle Theory -- 6.3.7.3 Zeros Near Fuchsian Singularities -- 6.3.8 Pseudo-Abelian Integrals -- 6.4 Many (Complex) Dimensions -- 6.4.1 Infinitesimal Version: Multiplicity ; Counting -- 6.4.1.1 Multiplicity of Maps in One Variable -- 6.4.1.2 Noetherian Multiplicities: The Isolated Case -- 6.4.1.3 Multiplicity Operators in the Multidimensional Case -- 6.4.1.4 Lower Bounds -- 6.4.1.5 Rolle Inequality for the Multiplicity Operators.; 6.4.1.6 Application to Noetherian Functions.