Handbook of Geometry and Topology of Singularities VII

Preface
Contents
Contributors
1 Jim Damon's Contributions to Singularity Theory and Its Applications
	Contents
	1.1 Singularity Theory: Bill Bruce, David Mond and Les Wilson
		1.1.1 Topological Stability
		1.1.2 Subgroups of Mather's Groups
		1.1.3 Moving on from Isolated Hypersurface Singularities
		1.1.4 Semi-Coherence
		1.1.5 Application of Singularity Theory to Bifurcation Theory
		1.1.6 Generic Properties of Singularity Theory to Solutions of Partial Differential Equations
	1.2 Medial Structures: Peter Giblin
		1.2.1 The Blum Medial Axis: Examples
		1.2.2 Radial Vector Fields and Shape Operators in the Blum Case
		1.2.3 Differential Geometry of the Boundary B
		1.2.4 A Note on Ridges on B
		1.2.5 Height Ridges and Other Special Curves
		1.2.6 Skeletal Structures
		1.2.7 ``Rigidity'' Questions for Deformations of Regions and Medial Axes
	1.3 Illuminated Surfaces: Peter Giblin
		1.3.1 The Mathematical Setup
		1.3.2 Comments on Enumerating the Various Cases
		1.3.3 The Further Role of Geometry
		1.3.4 Techniques Needed for the Classification
	1.4 Applications to Medical Image Computing: Stephen Pizer
		1.4.1 Introduction
		1.4.2 Segmentation and Height Ridges
		1.4.3 Skeletal Models
		1.4.4 Multiple Objects
		1.4.5 Generalized Cylinders
		1.4.6 Principal Component Analysis
		1.4.7 Conclusion
	1.5 Jim as a Colleague
	1.6 Jim Damon's Papers
		1.6.1 Doctoral Dissertations in Mathematics with Jim Damon as Advisor
		1.6.2 Computer Science Dissertations Co-Advised by Jim Damon
	References
2 Real Function Singularities and Their Bifurcation Sets
	Contents
	2.1 Introduction
	2.2 Main Definitions
		2.2.1 Basic Terminology
		2.2.2 Discriminant, Caustic, Maxwell Set
		2.2.3 Ghosts of Complex Bifurcations
		2.2.4 Boundary Singularities and Their Discriminants
		2.2.5 Remark on the Global Theory
	2.3 First Examples
		2.3.1 Pictures of Discriminants
		2.3.2 Caustics
		2.3.3 Maxwell Sets
	2.4 Classification of Real Function Singularities
		2.4.1 Simple Singularities
		2.4.2 Parabolic Singularities
		2.4.3 Simple Boundary Singularities and Their Versal Deformations
	2.5 Discriminants, Wave Fronts and Projective Duality
		2.5.1 Projective Duality
		2.5.2 Wave Fronts
	2.6 Wave Fronts in Integral Geometry and Hyperbolic PDEs
		2.6.1 Singularities of Volume Functions
		2.6.2 Boundary/Asymptotic Version of the Volume Function
		2.6.3 Hyperbolic PDEs
	2.7 Caustics in Optics
		2.7.1 Caustics as Focal Sets
		2.7.2 Lagrangian Singularities
	2.8 Complements of Discriminants of Simple and Parabolic Singularities
		2.8.1 Results of E. Looijenga
		2.8.2 Topological Invariant of Non-discriminant Components
		2.8.3 On 1-Cohomology of Complements of Discriminants
		2.8.4 Complements of Discriminants of Parabolic Singularities
		2.8.5 Case of Simple Boundary Singularities
	2.9 Complements of Caustics of Simple and Parabolic Function Singularities
		2.9.1 Numbers of Components
		2.9.2 1-Cohomology of Components
		2.9.3 Higher Homology
		2.9.4 Invariants of Complements of Caustics
		2.9.5 Examples
		2.9.6 Exceptional Orbits
	2.10 Algorithmic Enumeration of Topological Types of Morsifications of Real Function Singularities
		2.10.1 Concept
		2.10.2 Virtual Morsifications and Surgeries
		2.10.3 The Work of the Program in the Case of Simple Singularities
		2.10.4 Main and Reduced Programs
		2.10.5 What Fails for Not Simple Singularities and What Can Be Saved
		2.10.6 Is the Algorithm Finite and What to Do Otherwise?
		2.10.7 Two-Dimensional Features
		2.10.8 Example
	2.11 A Global Aspect: Lagrange Characteristic Classes
	2.12 Generalized Discriminants and Their Simplicial Resolutions
		2.12.1 Examples
		2.12.2 Simplicial Resolution of Discriminant Sets
	References
3 Perturbation Theory of Polynomials and Linear Operators
	Contents
	3.1 Introduction
	3.2 Perturbation Theory with Real Analytic Coefficients
		3.2.1 Rellich's Theorems
		3.2.2 A Proof of Theorem 3.2.2
			3.2.2.1 Tschirnhausen Form
			3.2.2.2 Splitting
		3.2.3 Multiparameter Case
			3.2.3.1 Abhyankar–Jung Theorem
			3.2.3.2 Proof of Theorem 3.2.9
		3.2.4 Perturbation of Normal Matrices
			3.2.4.1 Remarks on the Proofs
	3.3 Differentiable Roots of Hyperbolic Polynomials
		3.3.1 Bronshtein's Theorem
		3.3.2 Towards a Proof of Bronshtein's Theorem
			3.3.2.1 Splitting
			3.3.2.2 Glaeser's Inequality
			3.3.2.3 Interpolation Estimates
			3.3.2.4 The Key Argument
			3.3.2.5 Proof of Bronshtein's Theorem
		3.3.3 Sufficient Conditions for Cp Roots
			3.3.3.1 The Effect of Positive Local Minima
			3.3.3.2 A Regularity Class for Taking Radicals
			3.3.3.3 Finite Order of Contact
			3.3.3.4 Definability: No Oscillation
		3.3.4 Gårding Hyperbolic Polynomials
			3.3.4.1 Gårding Hyperbolic and Real Stable Polynomials
			3.3.4.2 Characteristic Roots
		3.3.5 Eigenvalues of Hermitian Matrices
		3.3.6 Singular Values
		3.3.7 Eigenvalues of Normal Matrices
	3.4 Regularity of the Roots in the General (Nonhyperbolic) Case
		3.4.1 The Case of Radicals
		3.4.2 Optimal Sobolev Regularity of the Roots
			3.4.2.1 Multivalued Sobolev Functions
		3.4.3 Absolute Continuity via Desingularization: Formulas for the Roots
		3.4.4 Multiparameter Case
			3.4.4.1 Bounded Variation of the Roots
	3.5 Lifting Maps Over Invariants of Group Representations
		3.5.1 A Reformulation of the Regularity Problem for Hyperbolic Polynomials
		3.5.2 Orthogonal Representations of Compact Lie Groups
			3.5.2.1 Differentiable Lifting
			3.5.2.2 Real Analytic Lifting
		3.5.3 The Main Tools
			3.5.3.1 Removing Fixed Points
			3.5.3.2 Dominant Invariant
			3.5.3.3 The Slice Theorem
			3.5.3.4 Orbit Type Stratification
		3.5.4 Examples and Applications
			3.5.4.1 Differentiable Eigenvalues of Real Symmetric Matrices
			3.5.4.2 Differentiable Decomposition of Nonnegative Functions into Sums of Squares
			3.5.4.3 Polynomials with Symmetries
		3.5.5 A Reformulation of the Regularity Problem for General Polynomials
		3.5.6 Representations of Linearly Reductive Groups
			3.5.6.1 Sobolev Lifting
			3.5.6.2 Analytic Lifting
		3.5.7 Some Remarks on the Proofs
	3.6 Applications
		3.6.1 Zero Sets of Smooth Functions
		3.6.2 Extension to the Optimal Transport Between Algebraic Hypersurfaces
	Appendix A: Function Spaces
		Hölder Spaces
		Lebesgue Spaces and Weak Lebesgue Spaces
		Sobolev Spaces
		Functions of Bounded Variation
	Appendix B: The Space of Hyperbolic Polynomials
	References
4 Frontal Singularities and Related Problems
	Contents
	4.1 Introduction
	4.2 Frontal Hypersurfaces
		4.2.1 Examples of Frontal Singularities
		4.2.2 Frontal Hypersurface Singularities
		4.2.3 Lagrangian and Legendrian Geometry
		4.2.4 Deformations of Integral Mappings
		4.2.5 Frontal Deformations and Frontal Stability
	4.3 General Frontals
		4.3.1 Examples of General Frontals
		4.3.2 General Frontal Singularities
		4.3.3 Stability of General Frontals
		4.3.4 An Algebraic Characterisation of Frontals
	4.4 Openings
		4.4.1 Jacobi Modules, Ramification Modules and Openings
		4.4.2 Versal Openings
		4.4.3 The Cases of Corank ≥2
	4.5 Other Topics
		4.5.1 Relation to Symplectic Geometry
		4.5.2 Frontal Jets
		4.5.3 Cofrontals
	4.6 Problems
		4.6.1 Problems on Frontals
		4.6.2 Problems on General Frontals
		4.6.3 Problems on Openings
		4.6.4 Problems on Other Topics
	References
5 Introduction to Global Singularity Theory of Differentiable Maps
	Contents
	5.1 Introduction
	5.2 Special Generic Maps
		5.2.1 Fold Singularities
		5.2.2 Reeb Spaces of Special Generic Maps
		5.2.3 Structure Theorem of Special Generic Maps
		5.2.4 Special Generic Maps on Homotopy Spheres
		5.2.5 Standard Special Generic Maps and Gromoll Filtrations
		5.2.6 Manifolds Admitting Special Generic Maps
	5.3 Cobordism of Generic Maps
		5.3.1 Notion of Cobordism for Singular Maps
		5.3.2 Cobordism of Morse Functions
		5.3.3 Singular Fibers
	5.4 Complexity of Generic Maps
		5.4.1 Complexity of Stable Maps
		5.4.2 Gromov's Result
		5.4.3 Simplicial Volume
		5.4.4 Amenable Stratification
		5.4.5 Localization Lemma
		5.4.6 D-Inequality for G(n-1)-Fibers
	References
6 Singularities of Functions: A Global Point of View
	Contents
	6.1 Introduction
	6.2 Topological/Cohomological Properties
		6.2.1 The Fibration Theorem
		6.2.2 Cohomological Tools
	6.3 Cohomologies Attached to a Pair (U,f)
		6.3.1 Various Expressions of the Singular Cohomology with Growth Conditions
		6.3.2 Singular Homology with Growth Conditions
		6.3.3 Algebraic de Rham Cohomology
		6.3.4 Irregular Mixed Hodge Theory
	6.4 Monodromy Properties of a Pair (U,f)
		6.4.1 Monodromy on Global Vanishing Cycles Attached to (U,f)
		6.4.2 The Stokes Filtration Attached to a Pair (U,f)
		6.4.3 The Notion of a Stokes-Filtered Local System
		6.4.4 Pairings and Stokes Matrices
		6.4.5 The Twisted de Rham Complex with an Algebraic Parameter
		6.4.6 The Case of a Formal Parameter
		6.4.7 Irregular Mixed Hodge Theory
		6.4.8 Other Cohomology Theories
	6.5 Tameness on Smooth Affine Varieties
		6.5.1 Cohomological Properties of Tame Functions
		6.5.2 The Spectrum at Infinity
		6.5.3 An Example: Laurent Polynomials
	References
7 Floer Theory, Arc Spaces and Singularities
	Contents
	7.1 Introduction
	7.2 Hypersurface Singularities
		7.2.1 Topology
		7.2.2 Contact Geometry
		7.2.3 Morse Homology
		7.2.4 Floer Cohomology of Symplectomorphisms
		7.2.5 Floer Cohomology of Monodromy Maps of Isolated Singularities
		7.2.6 A Spectral Sequence for Floer Cohomology of Singularities
		7.2.7 Spectral Sequences in Floer Theory
		7.2.8 Idea of Proof of Theorem 7.2.52
		7.2.9 The Work of de Bobadilla and Pelka
		7.2.10 Arc Spaces and Hypersurface Singularities
		7.2.11 Contact Loci and Floer Cohomology
		7.2.12 Further Speculations
	7.3 General Isolated Singularities
		7.3.1 Links of Isolated Singularities
		7.3.2 Minimal Discrepancy
		7.3.3 Full Contact Homology
		7.3.4 Full Contact Homology of Links
		7.3.5 A Spectral Sequence for Contact Homology
		7.3.6 The Arc Space of a General Isolated Singularity
	7.4 Quotient Singularities and the McKay Correspondence
		7.4.1 McKay Correspondence and Arc Spaces
		7.4.2 Floer Cohomology of Quotient Singularities
	References
8 Various Derivation Lie Algebras of Isolated Singularities
	Contents
	8.1 Introduction
	8.2 Derivation Lie Algebra
	8.3 Higher Nash Blow-Up Derivation Lie Algebra
	8.4 Higher Nash Blow-Up and Higher Order Jacobian Matrix
	8.5 Halperin Conjecture
	8.6 k-th Singular Locus Derivation Lie Algebra
	8.7 Hodge Moduli Algebra and Its Derivation Lie Algebra
	8.8 Fewnomial Singularities
	8.9 k-th Yau Algebra
	8.10 Hessian Derivation Lie Algebra
	8.11 Higher Order Moduli Algebra and Its Derivation Lie Algebra
	8.12 Torelli-Type Theorems
	8.13  Isolated Complete Intersection Singularities
	References
9 Three Dimensional Rational Isolated Complete Intersection Singularities
	Contents
	9.1 Introduction
	9.2 Preliminaries
		9.2.1 Complete Intersection and Gorenstein Singularities
		9.2.2 Rational and Elliptic Singularities
		9.2.3 Weighted Homogeneous Singularities
		9.2.4 Rational Weighted Homogeneous ICISs
	9.3 Embedding Dimension of Rational ICISs
	9.4 Three Dimensional Rational Hypersurface Singularities
	9.5 Three Dimensional Rational ICISs
	References
10 Stability of klt Singularities
	Contents
	10.1 Introduction
		10.1.1 Motivation
		10.1.2 History
		10.1.3 Outline
		10.1.4 Notation and Conventions
	10.2 Stable Degeneration
		10.2.1 Valuation
		10.2.2 Fano Cone Singularities
		10.2.3 Normalized Volume
		10.2.4 Stable Degeneration Conjecture
	10.3 Kollár Components
		10.3.1 Divisorial Minimizer
	10.4 Geometry of Minimizers
		10.4.1 Existence
		10.4.2 Uniqueness and K-Semistability
		10.4.3 Quasi-Monomial Property
		10.4.4 Finite Generation
		10.4.5 K-Polystable Degeneration
	10.5 Boundedness of Singularities
	10.6 Questions and Future Directions
		10.6.1 Boundedness
		10.6.2 Local Volumes
		10.6.3 Miscellaneous
	References
11 An Introduction to V-Filtrations
	Contents
	11.1 Introduction
	11.2 A Brief Review of D-Module Theory
		11.2.1 The Sheaf of Differential Operators
		11.2.2 The de Rham Complex
		11.2.3 The Left–Right Equivalence
		11.2.4 Singular Support and Holonomic D-Modules
		11.2.5 Functors on D-Modules
		11.2.6 The Riemann-Hilbert Correspondence
	11.3 The V-Filtration with Respect to a Smooth Hypersurface
		11.3.1 The Filtration V•DX
		11.3.2 The V-Filtration with Respect to t
		11.3.3 The Maps Var and can
	11.4 The V-Filtration with Respect to an Arbitrary Hypersurface
	11.5 V-Filtrations and b-Functions
		11.5.1 Basics on b-Functions
		11.5.2 Existence of the V-Filtration via b-Functions
	11.6 An Example: Weighted Homogeneous, Isolated Singularities
		11.6.1 The V-Filtration for Weighted Homogeneous Polynomials with Isolated Singularities
		11.6.2 The b-Function of Weighted Homogeneous Polynomials with Isolated Singularities
	11.7 Duality and V-Filtrations
	11.8 Push-Forward and V-Filtrations
		11.8.1 The Case of Smooth Projections
		11.8.2 The General Case
	11.9 The Comparison with the Topological Vanishing and Nearby Cycles
		11.9.1 Topological Vanishing and Nearby Cycles
		11.9.2 The Vanishing and Nearby Cycles via the V-Filtration
	11.10 The V-Filtration on Bf and Invariants of Singularities
		11.10.1 The Reduced Bernstein-Sato Polynomial
		11.10.2 The V-Filtration and Multiplier Ideals
		11.10.3 The V-Filtration and the Minimal Exponent
	References
12 Geometric Monodromies, Mixed Hodge Numbers of Motivic Milnor Fibers and Newton Polyhedra
	Contents
	12.1 Introduction
	12.2 Milnor Fibers and Nearby Cycle Sheaves
	12.3 Theorems of Kouchnirenko, Varchenko and Oka
	12.4 Singularities and Monodromies at Infinity of Polynomial Maps
	12.5 Combinatorial Formulas for Equivariant Hodge-Deligne Numbers of Toric Hypersurfaces
	12.6 Motivic Milnor Fibers and Their Hodge Realizations
	12.7 Motivic Milnor Fibers at Infinity and Their Applications
	12.8 Jordan Normal Forms of Milnor Monodromies
	12.9 Theory of Katz and Stapledon
		12.9.1 Equivariant Ehrhart Theory of Katz-Stapledon
		12.9.2 Stapledon's Full Combinatorial Description of the Jordan Normal Forms of Milnor Monodromies
	12.10 Milnor Fibers and Monodromies of Meromorphic Functions
	References
13 Introduction to the Monodromy Conjecture
	Contents
	13.1 Introduction
	13.2 History
		13.2.1 Archimedean Zeta Functions
		13.2.2 Relation with Monodromy
		13.2.3 p-Adic Zeta Functions
		13.2.4 The Conjecture; p-Adic Version
	13.3 More `Igusa Type' Zeta Functions and Formulas via Resolution
		13.3.1 Denef's Formula
		13.3.2 The Topological Zeta Function
		13.3.3 The Motivic Zeta Function
		13.3.4 Monodromy and b-Function
	13.4 The Case of Plane Curves (n=2)
		13.4.1 Non-contribution
		13.4.2 Proof of the Conjecture
		13.4.3 Structure of Resolution Graph and Determination of All Poles
	13.5 Higher Dimension
		13.5.1 Strategy for Surfaces (n=3)
		13.5.2 Special Polynomials
	13.6 Generalizations
		13.6.1 Original Formulation
		13.6.2 Mappings/Ideals
		13.6.3 Zeta Functions with Differential Form
		13.6.4 Multivariate Zeta Functions
	References
14 Applications of Singularity Theory in Applied Algebraic Geometry and Algebraic Statistics
	Contents
	14.1 Introduction
	14.2 Preliminaries
		14.2.1 Whitney Stratification
		14.2.2 Constructible Functions and Local Euler Obstruction
		14.2.3 Hypersurface Singularities, Milnor Fiber
		14.2.4 Nearby and Vanishing Cycle Functors
		14.2.5 Conormal Varieties: Characteristic Cycles
		14.2.6 Chern Classes of Singular Varieties
		14.2.7 Microlocal Interpretation of Chern Classes
		14.2.8 Chern Classes via Logarithmic Cotangent Bundles
	14.3 Nearest Point Problems: Euclidean Distance Degree
		14.3.1 Classical Examples of Nearest Point Problems
		14.3.2 ED Degrees of Complex Affine Varieties: Multiview Conjecture
			14.3.2.1 Euclidean Distance Degree
			14.3.2.2 Topological Interpretation of ED Degrees
			14.3.2.3 Multiview Conjecture
		14.3.3 Projective Euclidean Distance Degree
		14.3.4 Defect of ED Degree
		14.3.5 Other Developments
	14.4 Maximum Likelihood Estimation
		14.4.1 ML Degree of Very Affine Varieties
		14.4.2 Likelihood Geometry in CPn
			14.4.2.1 Sectional ML Degrees and the Involution Conjecture
		14.4.3 Other Developments
			14.4.3.1 ML Degree of Mixture of Two Independence Models
			14.4.3.2 Computing Local Euler Obstruction from Sectional ML Degrees
			14.4.3.3 ML Degree of a Sparse Polynomial System
			14.4.3.4 ML Data Discriminant for Positive Real Solutions
			14.4.3.5 Gaussian Models and Symmetric Matrices
	14.5 Linear Optimization on a Variety
		14.5.1 Linear Optimization Degree
		14.5.2 Linear Optimization Bidegrees and Chern-Mather Classes
		14.5.3 Sectional Linear Optimization Degrees: Relation to LO Bidegrees
		14.5.4 Relation to Polar Degrees
	14.6 Non-generic Data: Morsification and Applications
		14.6.1 Morsification
		14.6.2 Computing Multiplicities
	References
Index