Handbook of geometry and topology of singularities. II

Main subject categories: • Geometry • Topology • Singularities • Homology • Stratification theory • Complex surfaces • HypersurfacesThis is the second volume of the Handbook of the Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. This volume consists of ten chapters which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory and related topics.Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and in other subjects. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways.The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook. 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 10.2.1 Braid Monodromy Presentation of Fundamental Groups. 10.2.2 Abelian Fundamental Groups 10.3 Alexander Invariants 10.3.1 Alexander Polynomials 10.3.2 A Divisibility Theorem 10.3.3 Branched Covers 10.3.4 Abelian Covers 10.3.5 Characteristic Varieties 10.3.6 Isolated Non-normal Crossings 10.3.7 Twisted Alexander Invariants 10.3.8 Alexander Invariants of the Complements Without Isolatedness Properties 10.4 Ideals of Quasiadjunction and Multiplier Ideals 10.4.1 Ideals and Polytopes of Quasi-adjunction 10.4.2 Ideals of Quasi-adjunction and Multiplier Ideals 10.4.3 Local Polytopes of Quasi-adjunction and Spectrum of Singularities 10.4.4 Ideals of Quasi-adjunction and Homology of Branched Covers 10.4.5 Hodge Decomposition of Characteristic Varieties 10.4.6 Bernstein-Sato Ideals and Polytopes of Quasi-adjunction 10.5 Asymptotic of Invariants of Fundamental Groups 10.6 Special Curves 10.6.1 Arrangements of Lines, Hyperplanes and Plane Curves 10.6.2 Generic Projections 10.6.3 Complements to Discriminants of Universal Unfoldings 10.6.4 Complements to Discriminants of Complete Linear Systems 10.6.5 Plane Sextics and Trigonal Curves 10.6.6 Zariski Pairs References Index This is the second volume of the Handbook of the Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. This volume consists of ten chapters which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory and related topics. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and in other subjects. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.