Arrangements, Local Systems and Singularities (eBook)

Table of Contents 6
Preface 7
List of Participants 9
Combinatorics of Covers of Complexified Hyperplane Arrangements 11
Introduction 11
1. Definitions and background 13
1.1. Arrangements 13
1.2. Posets 14
1.3. The Salvetti complex 15
1.4. The arrangement groupoid 16
2. Homotopy colimits for combinatorial applications 21
3. Salvetti-type diagram models 23
3.1. Covering maps 25
3.2. The fundamental group 26
3.3. Classification of the covers 27
3.4. Reflection arrangements and Charney-Davis models 30
4. Paris’ topological models 31
4.1. Oriented systems and their covers 31
4.2. Topological covers associated to oriented systems 33
4.3. Combinatorial stratifications and diagram models 34
5. Garside-type diagram models 36
5.1. The ‘Strong Lattice Property’ 39
6. Applications and open ends 42
References 44
Homological Aspects of Hyperplane Arrangements 49
1. Introduction 49
1.1. Free resolutions 50
1.2. Fundamental groups and the lower central series 51
1.3. A third Lie algebra 52
2. Resolutions over the Orlik-Solomon algebra 54
2.1. Koszul algebras 54
2.2. Decomposable arrangements 55
2.3. The homotopy Lie algebra 56
3. Part II: Resolutions over the exterior algebra 58
3.1. The resolution of A over the exterior algebra 58
3.2. Homology of an arrangement complement 59
3.3. The resolution of A* over the exterior algebra 60
3.4. Koszul modules 62
3.5. Resonance 63
3.6. Resonance and Betti numbers 64
References 65
Pencils of Plane Curves and Characteristic Varieties 69
1. Introduction 69
2. On rational maps from P2 to P1 71
3. Characteristic varieties and resonance varieties 72
4. Minimal arrangements 80
5. Fibered complements and K(p, 1)-spaces 81
6. Translated components 83
References 91
Local Systems and Constructible Sheaves 121
Introduction 121
1. Local Systems 123
1.1. Background in undergraduate studies 123
1.2. Definition and properties 124
1.3. Local systems and Representations of the fundamental group 125
1.4. System of n linear first-order differential equations and local systems 128
1.5. Connections and Local Systems 129
1.6. Fibrations and local systems (Gauss-Manin connection) 132
2. Singularities of Local Systems and Systems of differential equations with meromorphic coefficients: Regularity 136
2.1. System with meromorphic coefficients on the complex disc 137
2.2. Connections with Logarithmic singularities 141
2.3. Meromorphic Connections on the disc 145
2.4. Regular meromorphic connections 146
3. Singularities of local systems: Constructible Sheaves 148
3.1. Stratification theory 149
3.2. Cohomologically Constructible sheaves 152
4. From Lefschetz theorems to the decomposition theorem 154
4.1. The decomposition theorem for projective morphisms 155
References 157
Geometry and Combinatorics of Resonant Weights 164
1. Arrangements and projective arrangements 165
2. Master functions and local systems 166
3. Euler characteristics and Hilbert series 169
4. Resonance varieties 171
5. Critical loci of master functions 180
References 183
The Characteristic Quasi-Polynomials of the Arrangements of Root Systems and Mid-Hyperplane Arrangements 186
1. Introduction 186
2. Results on the characteristic quasi-polynomial of an integral matrix 188
3. Arrangements of root systems 192
4. Mid-hyperplane arrangement 197
References 198
Toric Varieties and the Diagonal Property 200
1. Introduction 200
2. Toric Varieties 201
2.1. Definition and examples 201
2.2. Affine toric varieties 202
2.3. Fans 205
3. The Diagonal Property 209
4. The Diagonal Property on Toric Surfaces 211
References 215
Introduction to Plane Curve Singularities. Toric Resolution Tower and Puiseux Pairs 217
1. Introduction 217
2. Preliminaries 217
2.1. Presentations of a germ of a curve 217
2.2. Intersection number 218
2.3. Newton boundary 219
2.4. Weierstrass preparation theorem 220
2.5. Resultant and discriminant 221
2.6. Weierstrass polynomial and discriminant 221
2.7. Puiseux expansion 222
3. Resolution of Singularities 223
3.1. Blowing-up 223
3.2. Tangent cone 225
3.3. Resolution of Singularities 226
3.4. Geometry of the resolution 227
3.5. Toric modification 228
3.6. Non-degeneracy and a canonical resolution 231
3.7. Relation between a tower of ordinary blowing-ups and a toric blowing-up 236
4. Milnor fibration 237
4.1. Milnor fibration 237
4.2. Non-degenerate plane curves 239
5. Puiseux characteristics and their geometry 240
5.1. Change of parameters 240
5.2. Tower of toric modifications and Puiseux characteristics 244
5.3. Toric resolution tower from a Puiseux expansion 245
5.4. Resolution graph and the zeta function of the Milnor fibration 248
References 253
Surface Singularities Appeared in the Hyperbolic Schwarz Map for the Hypergeometric Equation 254
1. Introduction 254
2. Swallowtail as a discriminant 256
2.1. Discriminant of a quadratic polynomial 256
2.2. Discriminant of a cubic plynomial 257
2.3. Discriminant of a quartic polynomial 259
2.4. Parametrization of the surface S 261
3. Hypergeometric differential equation 262
4. The Schwarz map of the hypergeometric differential equation 265
4.1. Schwarz triangle when exponents are real 265
4.2. Schwarz map when exponents are real 266
4.3. Schwarz map when the exponents are not real 268
5. Hyperbolic spaces 268
5.1. 2-dimensional models 268
5.2. 3-dimensional models 268
6. Hyperbolic Schwarz map – fundamental properties 269
6.1. Monodromy group 270
6.2. Use of the Schwarz map 270
6.3. Singularities of hyperbolic Schwarz maps 271
7. Hyperbolic Schwarz maps – Examples 272
7.1. Finite (polyhedral) monodromy groups 272
7.2. A Fuchsian monodromy group 274
7.3. Parallel family of flat fronts connecting Schwarz and derived Schwarz maps 275
7.4. Confluence of swallowtail singularities 276
References 278
On the Extendability of Free Multiarrangements 280
1. Introduction 280
2. Extendability of locally A2 arrangements 282
3. Proof 284
4. Totally non-free arrangements 285
5. Free interpolations between extended Shi and Catalan arrangements 285
References 287
Problem Session 289
1. Introduction 289
2. Problems 289
2.1. Ae-versal unfoldings of differentiable map germs 289
2.2. Multiple point spaces for maps 296
2.3. Invariance of the topological type 297
2.4. Representations of quivers 297
2.5. Injective maps 299
2.6. Representations of Lie Groups 300
2.7. Homotopy type of hyperplane arrangements 301
2.8. Resonance varieties 304
2.9. Torsion in the first fundamental group 309
2.10. Torsion in the homology of the Milnor fibre 311
2.11. Stability of hyperplane arrangements 312
2.12. Profinite completion of fundamental groups 315
References 316