This volume arises from the Fifth Franco-Japanese Symposium on Singularities, held in Strasbourg in August 2009. The conference brought together an international group of researchers, mainly from France and Japan, working on singularities in algebraic geometry, analytic geometry and topology. The conference also featured the JSPS Forum on Singularities and Applications, which aimed to introduce some recent applications of singularity theory to physics and statistics. This book contains research papers and short lecture notes on advanced topics on singularities. Some surveys on applications that were presented at the JSPS Forum are also included. Among the topics covered are splice surface singularities, b-functions, equisingularity, degenerating families of Riemann surfaces, hyperplane arrangements, mixed singularities, jet schemes, noncommutative blow-ups, characteristic classes of singular spaces, and applications to geometric optics, cosmology, and learning theory. Graduate students who wish to learn about various approaches to singularities, as well as experts in the field and researchers in other areas of mathematics and science, will find the contributions to this volume a rich source for further study and research
Author(s): Vincent Blanloeil, Toru Ohmoto
Series: Irma Lectures in Mathematics and Theoretical Physics
Publisher: European Mathematical Society
Year: 2012
Language: English
Pages: 370
Preface......Page 5
Contents......Page 7
A. Joets: Optical caustics and their modelling as singularities......Page 9
Observation of caustics......Page 10
First modellings......Page 12
Caustics as singularities of maps......Page 14
Caustics as Lagrangian singularities......Page 15
Caustics and wave front singularities......Page 16
Local types......Page 17
Global aspects......Page 19
Chekanov's formula for the singular set Sigma......Page 20
Topological formula for caustics......Page 22
References......Page 24
Introduction......Page 27
Results......Page 30
The use of vanishing cycles......Page 33
An auxiliary result......Page 37
Proof of Theorem 2.1 and Theorem 2.2......Page 39
References......Page 44
Introduction......Page 47
Preliminaries on jet schemes and positive characteristic methods......Page 48
Singularities of the jet schemes......Page 51
References......Page 57
Introduction: A very brief review of general relativity......Page 59
Geodesic incompleteness......Page 61
The endpoint set of the event horizon......Page 63
Apparent horizons......Page 66
Critical behaviour in gravitational collapse......Page 68
The phenomena......Page 69
Renormalisation group method......Page 71
The mechanism......Page 72
Topological singularities due to quotienting......Page 74
References......Page 76
Contents......Page 79
Main theorem......Page 80
Complex orbifolds......Page 84
Orbifold maps......Page 85
Orbifold pull-back diagram......Page 87
Types of mapping classes......Page 90
Fenchel–Nielsen coordinates......Page 92
Compactification process of M(Sigma_g)......Page 93
Bers' deformation spaces......Page 94
Subdeformation spaces D_epsilon(C)......Page 98
The universal degenerating family......Page 101
References......Page 108
Introduction......Page 111
Algebraic local cohomology attached to semiquasihomogeneous singularity......Page 112
b-function......Page 113
Computation of b-function......Page 115
Semiquasihomogeneous singularities with L(f)=2......Page 117
References......Page 123
Schemes......Page 125
Algebraic spaces......Page 126
Quotient stacks......Page 127
Chow group and pushforward......Page 128
Chern–MacPherson transformation......Page 131
Degree......Page 133
Deligne–Mumford stacks......Page 134
Modified pushforwards......Page 135
Other characteristic classes......Page 136
References......Page 138
Introduction......Page 141
Canonical orientation......Page 143
Milnor fiber......Page 144
Topology of mixed projective hypersurface......Page 145
Isolated singularity case......Page 146
Solutions and points in CP^1......Page 147
Degree of mixed projective hypersurfaces......Page 148
Milnor fibers......Page 150
Projective mixed curves......Page 152
References......Page 154
Introduction......Page 157
Splice quotients......Page 158
Filtrations......Page 160
Some analytic invariants......Page 161
The Seiberg–Witten invariant......Page 162
The multiplicity......Page 163
The embedding dimension......Page 165
References......Page 166
Introduction......Page 169
Generators for N_{n,q} and Hirzebruch–Jung continued fractions......Page 171
Duality of the singularities A_{n,q} and A_{n,n-q}......Page 173
Quasi-determinantal equations and the Artin component......Page 174
The grand monodromy covering......Page 176
Duality of the Artin components......Page 177
The case A_1......Page 178
The hypersurface case......Page 180
The case A_{n,1}......Page 183
The general case......Page 185
References......Page 187
Introduction......Page 189
Virtual classes of local complete intersections......Page 190
Functorial characteristic classes of singular spaces......Page 195
Nearby and vanishing cycles......Page 200
References......Page 211
Introduction......Page 215
Non-singular distributions......Page 217
Singular distributions......Page 218
Singular distributions on singular varieties......Page 219
Chern–Weil theory for virtual bundles......Page 220
Characteristic classes in the Cech–de Rham cohomology......Page 223
Local Chern classes and characters......Page 225
Riemann–Roch theorem for embeddings......Page 228
Localization by rank reason......Page 229
Residues and the local Chern classes......Page 231
An example......Page 233
Atiyah classes......Page 236
Cech–Dolbeault cohomology......Page 238
Relative Cech–Dolbeault cohomology......Page 240
Atiyah classes in Cech–Dolbeault cohomology......Page 241
Actions of distributions......Page 242
Localization and residues......Page 244
Atiyah classes on singular varieties......Page 245
An example......Page 248
References......Page 253
Introduction......Page 257
Preparation......Page 260
Real log canonical threshold......Page 262
Singular fluctuation......Page 263
Singular fluctuation......Page 269
Singular and realizable case......Page 270
Singular and unrealizable case......Page 271
Background of the problem......Page 272
Conclusion......Page 273
References......Page 274
Introduction......Page 277
Convention......Page 278
Pseudo-schemes......Page 279
Alterations and noncommutative blowups......Page 280
Equivalent modules......Page 282
F-steady modules and Frobenius morphisms......Page 283
Compatibility of Frobenius morphisms......Page 284
D-blowups......Page 287
Comparing Frobenius morphisms of commutative and noncommutative blowups......Page 288
References......Page 290
Introduction......Page 293
Counting: from cardinality to Hodge–Deligne polynomials......Page 296
An additive homology class......Page 299
A natural transformation associated to an additive homology class......Page 300
Examples......Page 306
Motivic characteristic classes: the most sophisticated categorification of additive-multiplicative homology classes......Page 312
Fulton–MacPherson's bivariant theory......Page 314
A pre-motivic bivariant theory on the category of complex algebraic varieties......Page 319
A bivariant-theoretic relative Grothendieck group K_{0}(V/X –f–> Y)......Page 324
A bivariant-theoretic motivic Hirzebruch class......Page 327
A zeta function of the motivic Hirzebruch class......Page 328
Some zeta functions......Page 329
Covariant functors and their zeta functions......Page 331
Natural transformations and their zeta functions......Page 340
A zeta function of the motivic Hirzebruch class......Page 343
References......Page 347
Minimality of hyperplane arrangements......Page 353
Non-resonant local systems......Page 354
Involution on unbounded chambers......Page 356
Generic flags......Page 358
Minimal complex arising from Lefschetz's Theorem......Page 360
The case =2......Page 362
An application......Page 366
Remarks and conjectures......Page 368
References......Page 369
