Handbook of geometry and topology of singularities II / / José Luis Cisneros-Molina, Dũng Tráng Lê, José Seado, editors

• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (581 pages)
• Disciplina: 516.35
• Soggetto topico: Singularities (Mathematics); Geometry, Algebraic; Topological groups; Singularitats (Matemàtica)
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): Cisneros MolinaJosé Luis; LêDũng Tráng; SeadoJosé
• Nota di contenuto: Intro -- Preface -- Contents -- Contributors -- 1 The Analytic Classification of Irreducible Plane Curve Singularities -- 1.1 Background -- 1.1.1 Plane Curve Singularities -- 1.1.2 Irreducible Plane Curve Singularities -- 1.1.3 Equisingularity of Branches -- 1.1.4 Semiroots of a Branch -- 1.2 Zariski's Approach -- 1.2.1 A Parameter Space -- 1.2.2 Kähler Differentials -- 1.2.3 The Zariski Invariant -- 1.3 Singularity Theory Approach -- 1.3.1 The Complete Transversal Theorem -- 1.3.2 Tangent Spaces to Orbits -- 1.3.3 The Analytic Classification -- 1.4 Final Remarks -- 1.4.1 Comparison with Other Works -- 1.4.2 Computability -- 1.4.3 A Solution for the Moduli Problem -- 1.4.4 Dimensions of Components of the Moduli Space -- 1.4.5 An Example -- 1.4.6 Analytic Versus Formal -- References -- 2 Plane Algebraic Curves with Prescribed Singularities -- 2.1 Introduction -- 2.1.1 Preliminaries: Isolated Singularities -- 2.2 Singular Plane Curves: Restrictions -- 2.2.1 Genus Formula and Bézout's Theorem -- 2.2.2 Plücker Formulae -- 2.2.3 Log-Miyaoka-Yau Inequality -- 2.2.4 Spectral Bound -- 2.3 Plane Curves with Nodes and Cusps -- 2.3.1 Plane Curves with Nodes -- 2.3.2 Plane Curves with Nodes and Cusps -- 2.4 Plane Curves with Arbitrary Singularities -- 2.4.1 Curves of Small Degrees -- 2.4.2 Curves with Simple, Ordinary, and Semi-quasihomogeneous Singularities -- 2.4.3 Curves with Arbitrary Singularities -- 2.5 Related and Open Problems -- 2.5.1 Existence Versus T-Smoothness and Irreducibility -- 2.5.2 Curves on Other Algebraic Surfaces -- 2.5.3 Other Related Problems -- 2.5.4 Some Questions and Conjectures -- References -- 3 Limit of Tangents on Complex Surfaces -- 3.1 Introduction -- 3.2 An Application of a Theorem of Hironaka -- 3.2.1 The Thom Stratification -- 3.2.2 Deformation on the Tangent Cone -- 3.2.3 Proof of Corollary 3.2.3. ; 3.3 The Theorem of Teissier -- 3.3.1 Statement -- 3.4 Hypersurfaces of Dimension 2 -- 3.4.1 Consequences of Teissier's Theorem -- 3.4.2 Limit of Tangents of Surfaces of mathbbC3 with Isolated Singularity -- 3.5 Polar Varieties of a Hypersurface of Dimension 2 -- 3.5.1 Polar Varieties -- 3.5.2 Exceptional Tangents of a Hypersurface of Dimension 2 -- 3.6 Surfaces in CN -- 3.6.1 Description of the Limits -- 3.6.2 Polar Curves -- 3.6.3 Relation with Discriminants of Projections to mathbbC2 -- 3.6.4 Exceptional Tangents and Equisingularity -- 3.6.5 Surfaces Without Exceptional Tangents -- 3.7 Appendix: Intersections in Grassmannians -- References -- 4 Algebro-Geometric Equisingularity of Zariski -- 4.1 Introduction -- 4.2 Equisingular Families of Plane Curve Singularities -- 4.2.1 Equisingular Families of Plane Curve Singularities. Definition -- 4.2.2 Equisingular Families of Plane Curve Singularities and Puiseux with Parameter -- 4.3 Zariski Equisingularity in Families -- 4.3.1 Topological Equisingularity and Topological Triviality -- 4.3.2 Arc-Wise Analytic Triviality -- 4.3.3 Whitney Fibering Conjecture -- 4.3.4 Algebraic Case -- 4.3.5 Principle of Generic Topological Equisingularity -- 4.3.6 Zariski's Theorem on the Fundamental Group -- 4.3.7 General Position Theorem -- 4.4 Construction of Equisingular Deformations -- 4.4.1 Global Polynomial Case -- 4.4.2 Application: Algebraic Sets are Homeomorphic to Algebraic Sets Defined Over Algebraic Number Fields -- 4.4.3 Analytic Case -- 4.4.4 Application: Analytic Set Germs are Homeomorphic to Algebraic Ones -- 4.4.5 Equisingularity of Function Germs -- 4.4.6 Local Topological Classification of Smooth Mappings -- 4.5 Equisingularity Along a Nonsingular Subspace. Zariski's Dimensionality Type -- 4.5.1 Equimultiplicity. Transversality of Projection. ; 4.5.2 Relation to Other Equisingularity Conditions. Examples -- 4.5.3 Lipschitz Equisingularity -- 4.5.4 Zariski Dimensionality Type. Motivation -- 4.5.5 Zariski Dimensionality Type -- 4.5.6 Almost all Projections -- 4.5.7 Canonical Stratification of Hypersurfaces -- 4.5.8 Zariski Equisingularity and Equiresolution of Singularities -- 4.6 Appendix. Generalized Discriminants -- References -- 5 Intersection Homology -- 5.1 Introduction -- 5.2 Classical Results-Poincaré and Poincaré-Lefchetz -- 5.2.1 PL-Structures -- 5.2.2 Pseudomanifolds -- 5.2.3 Stratifications -- 5.2.4 Borel-Moore Homology -- 5.2.5 Poincaré Duality Homomorphism -- 5.2.6 Poincaré-Lefschetz Homomorphism -- 5.3 The Useful Tools: Sheaves-Derived Category -- 5.3.1 Sheaves -- 5.3.2 System of Local Coefficients -- 5.3.3 Complexes of Sheaves -- 5.3.4 Injective Resolutions -- 5.3.5 Hypercohomology -- 5.3.6 The (Constructible) Derived Category -- 5.3.7 Derived Functors -- 5.3.8 Dualizing Complex -- 5.4 Intersection Homology-Geometric and Sheaf Definitions -- 5.4.1 The Definition for PL-Stratified Pseudomanifolds -- 5.4.2 Definition with Local Systems -- 5.4.3 Witt Spaces -- 5.4.4 The Intersection Homology Sheaf Complex -- 5.4.5 The Deligne Construction -- 5.4.6 Local Calculus and Consequences -- 5.4.7 Characterizations of the Intersection Complex -- 5.5 Main Properties of Intersection Homology -- 5.5.1 First Properties -- 5.5.2 Functoriality -- 5.5.3 Lefschetz Fixed Points and Coincidence Theorems -- 5.5.4 Morse Theory -- 5.5.5 De Rham Theorems -- 5.5.6 Steenrod Squares, Cobordism and Wu Classes -- 5.6 Supplement: More Applications and Developments -- 5.6.1 Toric Varieties -- 5.6.2 The Asymptotic Set -- 5.6.3 Factorization of Poincaré Morphism for Toric Varieties -- 5.6.4 General Perversities -- 5.6.5 Equivariant Intersection Cohomology -- 5.6.6 Intersection Spaces. ; 5.6.7 Blown-Up Intersection Homology -- 5.6.8 Real Intersection Homology -- 5.6.9 Perverse Sheaves and Applications -- References -- 6 Milnor's Fibration Theorem for Real and Complex Singularities -- 6.1 Introduction -- 6.2 Exotic Spheres and the Birth of Milnor's Fibration -- 6.2.1 Singularities and Exotic Spheres -- 6.2.2 Open Questions -- 6.3 Model Example: the Brieskorn-Pham Singularities -- 6.3.1 Weighted Homogeneous Singularities -- 6.3.2 Real Analytic Singularities -- 6.4 Local Conical Structure of Analytic Sets -- 6.5 The Classical Fibration Theorems for Complex Singularities -- 6.6 Topology of the Link and the Fiber -- 6.6.1 The Link -- 6.6.2 The Fiber -- 6.6.3 Vanishing Cycles, Open-Books and the Monodromy -- 6.7 Extensions and Refinements of Milnor's Fibration Theorem -- 6.8 Milnor Fibration for Real Analytic Maps -- 6.8.1 Strong Milnor Condition -- 6.8.2 Model Singularities -- 6.9 On Functions with a Non-isolated Critical Point -- 6.9.1 Functions with an Isolated Critical Value -- 6.9.2 Polar Weighted Singularities -- 6.9.3 Functions with Arbitrary Discriminant -- 6.10 Milnor Fibrations and d-Regularity -- 6.10.1 The Case of an Isolated Critical Value -- 6.10.2 The General Case -- 6.11 Singularities of Mixed Functions -- References -- 7 Lê Cycles and Numbers of Hypersurface Singularities -- 7.1 Introduction and Earlier Results -- 7.2 Definitions and Basic Properties of Lê Cycles and Numbers -- 7.3 Lê Numbers and the Topology of the Milnor Fiber -- 7.4 Lê-Iomdine Formulas and Thom's Af Condition -- 7.5 Aligned Singularities and Hyperplane Arrangements -- 7.6 Other Characterizations of the Lê Cycles -- 7.7 Projective Lê Cycles -- References -- 8 Introduction to Mixed Hypersurface Singularity -- 8.1 A Quick Trip to the Complex Hypersurface Singularity Theory -- 8.1.1 Milnor Fibration -- 8.1.2 The Hamm-Lê lemma and a Tubular Milnor Fibration. ; 8.1.3 Weighted Homogeneous Polynomials -- 8.1.4 Newton Boundary and Non-degeneracy -- 8.2 Mixed Hypersurface Singularities -- 8.2.1 Mixed Analytic Functions -- 8.2.2 Mixed Singularities -- 8.2.3 A Tubular Milnor Fibration of a Real Analytic Mapping -- 8.2.4 Stratification and Thom's af-Regularity -- 8.3 Milnor Fibrations for Mixed Functions -- 8.3.1 Mixed Functions and Newton Boundary -- 8.3.2 Non-degeneracy of Mixed Functions -- 8.3.3 Mixed Functions of one Variable (n=1) -- 8.3.4 Mixed Weighted Homogeneous Polynomials -- 8.3.5 Milnor Fibrations for Strongly Non-degenerate Mixed Functions -- 8.3.6 The Milnor Fibration for Convenient Mixed Functions -- 8.3.7 The Spherical Milnor Fibration -- 8.3.8 Milnor Fibrations for Non-convenient Mixed Functions -- 8.3.9 Topological Stability -- 8.3.10 Equivalence of Tubular and Spherical Milnor Fibrations -- 8.3.11 Real Blowing Up and a Resolution of a Real Type -- 8.3.12 Simplicial Mixed Polynomials -- 8.3.13 The Join Theorem -- 8.3.14 Topology of the Milnor Fiber -- 8.3.15 The Milnor Fibration for fbarg -- 8.3.16 Mixed Projective Hypersurfaces -- 8.3.17 Remarks and Problems -- References -- 9 From Singularities to Polyhedral Products -- 9.1 Introduction -- 9.2 Singularity Theory -- 9.3 Dynamical Systems -- 9.3.1 Complex Differential Equations -- 9.3.2 Higher Dimensional Group Actions -- 9.3.3 Generalized Hopf Bifurcations -- 9.4 Geometry -- 9.4.1 Complex Geometry -- 9.4.2 Contact and Symplectic Geometry -- 9.5 To the Polyhedral Product Functor -- 9.5.1 Coxeter Groups, Small Covers and Toric Manifolds -- 9.5.2 The Polyhedral Product Functor -- 9.6 Back to Singularity Theory -- 9.6.1 Quadratic Cones -- 9.6.2 Singular Intersections and Smoothings -- References -- 10 Complements to Ample Divisors and Singularities -- 10.1 Introduction. ; 10.2 Braid Monodromy, Presentations of Fundamental Groups and Sufficient Conditions for Commutativity.
• Titolo autorizzato: Handbook of geometry and topology of singularities II
• ISBN: 3-030-78024-4
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466405403316
• Lo trovi qui: Univ. di Salerno