Singularities and Their Interaction with Geometry and Low Dimensional Topology: In Honor of András Némethi (Trends in Mathematics)

The book is a collection of surveys and original research articles concentrating on new perspectives and research directions at the crossroads of algebraic geometry, topology, and singularity theory. The papers, written by leading researchers working on various topics of the above fields, are the outcome of the “Némethi60: Geometry and Topology of Singularities” conference held at the Alfréd Rényi Institute of Mathematics in Budapest, from May 27 to 31, 2019. Both the conference and this resulting volume are in honor of Professor András Némethi, on the occasion of his 60th birthday, whose work plays a decisive and influential role in the interactions between the above fields. The book should serve as a valuable resource for graduate students and researchers to deepen the new perspectives, methods, and connections between geometry and topology regarding singularities. Preface The Scientific Life and Work of András Némethi Contents Versality, Bounds of Global Tjurina Numbers and Logarithmic Vector Fields Along Hypersurfaces with Isolated Singularities 1 Introduction 2 Versality of Hypersurfaces with Isolated Singularities 3 Bounds on the Global Tjurina Number, Stability and Torelli Properties References On Ideal Filtrations for Newton Nondegenerate Surface Singularities 1 Introduction 2 Associated Power Series and the Search for an Equation 3 Newton Nondegeneracy 4 The Intersection Lattice 5 Cycles, Newton Diagrams and the Cone 6 Equality Between Ideals 7 Suspension Singularities 8 An Example References Young Walls and Equivariant Hilbert Schemes of Points in Type D 1 Introduction 2 Young Walls of Type Dn 3 Abacus Combinatorics 4 Enumeration of Young Walls References Real Seifert Forms, Hodge Numbers and Blanchfield Pairings 1 Introduction 2 Milnor Fibration and Picard-Lefschetz Theory 3 Hermitian Variation Structures and Their Classification 3.1 Abstract Definition 3.2 Classification of HVS Over C 3.3 The Mod 2 Spectrum 4 HVS for Knots and Links 4.1 Three Results of Keef 4.2 HVS for Links and Classical Invariants 4.3 Signatures, HVS and Semicontinuity of the Spectrum 5 Blanchfield Forms 5.1 Definitions 5.2 Blanchfield Pairing Over R[t,t-1] 5.3 Variation Operators and Linking Forms 6 Twisted Blanchfield Forms and Applications 6.1 Construction of Twisted Pairings 6.2 Twisted Hodge Numbers and Twisted Signatures 6.3 A Few Words on Case F=C 6.4 A Closing Remark References ħ-Deformed Schubert Calculus in Equivariant Cohomology, K-Theory, and Elliptic Cohomology 1 Introduction 2 Ordinary and Equivariant Cohomological Schubert Calculus 2.1 Schubert Classes and Structure Coefficients 3 The Main Theorem 4 The Partial Flag Variety 5 Elliptic Functions 5.1 Theta Functions 5.2 Fay's Trisecant Identity 6 Equivariant Elliptic Cohomology of Fλ 7 ħ-Deformed Schubert Classes in H*, K, and Ell 7.1 ħ-Deformed Schubert Class in Cohomology: CSM Class 7.2 ħ-Deformed Schubert Class in K Theory: Motivic Chern Class 7.3 ħ-Deformed Schubert Class in Elliptic Cohomology: The Elliptic Class 8 Weight Functions and Their Orthogonality Relations 8.1 Rational Weight Functions 8.2 Trigonometric Weight Functions 8.3 Elliptic Weight Functions 8.4 Orthogonality 8.5 Weight Functions Represent ħ-Deformed Schubert Classes 9 Sample Schubert Structure Constants 9.1 Cohomology 9.2 Equivariant Elliptic Cohomology 9.3 Non-equivariant Elliptic Cohomology 9.4 Positivity? References Fundamental Groups and Path Lifting for Algebraic Varieties 1 Introduction 2 Maps Between Fundamental Groups 3 Open and Universally Open Maps 4 Path Lifting in the Euclidean Topology 4.1 Examples References Cremona Transformations of Weighted Projective Planes, Zariski Pairs, and Rational Cuspidal Curves 1 Introduction 2 Quotient Singularities and Weighted Cremona Transformations 2.1 Curves in Quotient Surface Singularities 2.2 Weighted Projective Planes 2.3 Weighted Blow-ups 2.4 Weighted Cremona Transformations 3 Zariski Pairs on Weighted Projective Planes 3.1 Fundamental Groups of Complements 3.2 A Family of Zariski Pairs of Irreducible Weighted Projective Curves 3.3 Cyclic Covers and Their Irregularity à la Esnault–Viehweg 4 Some Rational Cuspidal Curves on Weighted Projective Planes 4.1 Rational Cuspidal Curves via Weighted Cremona Transformations 4.2 Rational Cuspidal Curves via Weighted Kummer Covers 4.3 A rational Cuspidal Curve with Four Cusps 4.4 Milnor Fibers 5 Weighted Lê–Yomdin Surface Singularities 5.1 The Determinant of a Normal Surface Singularity 5.2 Superisolated and Lê–Yomdin Singularities 5.3 Weighted Lê–Yomdin Singularities 6 Normal Surface Singularities with Rational Homology Sphere Links 6.1 Brieskorn–Pham Singularities 6.2 Examples from Cremona and Kummer 6.3 Integral Homology Sphere Surface Singularities References Normal Reduction Numbers of Normal Surface Singularities 1 Introduction 2 Cycles and Cohomology 3 Cohomology and Normal Reduction Numbers 4 Cone-Like Singularities 4.1 Homogeneous Hypersurface Singularities 5 Brieskorn Complete Intersections 5.1 The Maximal Ideal Cycle, the Fundamental Cycle, and the Canonical Cycle 5.2 The Normal Reduction Numbers 5.3 Elliptic Singularities of Brieskorn Type References Motivic Chern Classes of Cones 1 Introduction 2 What Is the K-Class of a Subvariety? 2.1 Algebraic K-Theory and the Sheaf K-Class 2.2 Topological K-Theory 2.3 The K-Theory of Pn 2.4 Hilbert Polynomial 2.5 The Pushforward K-Class 2.6 Motivic Invariants and the Motivic K-Class 2.7 The Todd Genus 2.8 Genus of Smooth Hypersurfaces 2.9 The χy Genus 3 Cones 3.1 The Projective Case 4 Equivariant Classes 4.1 Universal Classes in K-Theory 4.2 Equivariant Classes of Cones in Cohomology: The Projective Thom Polynomial 4.3 Projective Thom Polynomial for the Motivic Chern Class 4.3.1 The Kirwan Map in K-Theory 4.3.2 The Affine to Projective Formula 4.4 The Projective to Affine Formula References Semicontinuity of Singularity Invariants in Families of Formal Power Series 1 Introduction 2 Quasi-Finite Modules and Semicontinuity 2.1 The Completed Tensor Product 2.2 Fibre and Completed Fibre 2.3 Semicontinuity Over a 1-Dimensional Ring 2.4 Henselian Rings and Henselian Tensor Product 2.5 Semicontinuity for Algebraically Presented Modules 2.6 Related Results 3 Singularity Invariants 3.1 Isolated Singularities and Flatness 3.2 Milnor Number and Tjurina Number of Hypersurface Singularities 3.3 Determinacy of Ideals 3.4 Tjurina Number of Complete Intersection Singularities References Lattices and Correction Terms 1 Introduction 2 Lattices 3 Rational Homology Spheres and Correction Terms 4 Examples References Complex Surface Singularities with Rational HomologyDisk Smoothings 1 Introduction 2 Locations of -1 Curves 3 How to Find Sets of -1 Curves 4 Type Γ=W(p,q,r) 5 Type Γ=N(p,q,r), p>0 5.1 Case I for N(p,q,r), p>0 5.2 Case II for N(p,q,r), p>0 5.3 Case III for N(p,q,r), p>0 5.4 Type Γ=N(0,q,r) 6 Type Γ=M(p,q,r) 6.1 Case III for M(p,q,r), p,r>0 6.2 Case II for M(p,q,r), p,r>0 6.3 Case I for M(p,q,r), p,r>0 6.4 Type Γ=M(0,q,r), r ≥1 6.5 Type Γ=M(p,q,0), p≥1 6.6 Type Γ=M(0,q,0) 7 Self-Isotropic Subgroups and Fowler's Method 7.1 Fowler's Approach 7.2 Number of QHD Smoothing Components Appendix References On Tjurina Transform and Resolution of Determinantal Singularities 1 Introduction 2 Preliminaries 2.1 Determinantal Singularities 2.2 Transformations 3 Resolutions of the Model Determinantal Varieties 4 Transformations of General Determinantal Singularities 5 When Is the Tjurina Transform a Complete Intersection 6 Using Tjurina Transform to Resolve Hypersurface Singularities References On the Boundary of the Milnor Fiber 1 Introduction 2 From the Non-critical Level to the Special Fiber 3 The Case of Complex Surfaces 3.1 A Glance on Némethi-Szilárd's Work for Surface Singularities 3.2 On the Work of Michel-Pichon-Weber 3.3 On the Work of Fernández de Bobadilla and Menegon 4 The Vanishing Zone 5 The Vanishing Boundary Homology 6 Concluding Remarks References