SINGULARITY THEORY Dedicated to Jean-Paul Brasselet on His 60th Birthday Proceedings of the 2005 Marseille Singularity School and Conference CIRM, Marseille, France, 24 January – 25 February 2005

Introduction......Page 14
Part I: Five Courses on Singularity Theory......Page 18
1. Introduction......Page 20
3: Obstruction theory - The smooth case......Page 23
4.1. Stratifications, triangulations and cell decompositions......Page 27
4.2.1. Parallel Extension......Page 29
4.2.2. Transversal vector field......Page 30
4.3. Chern classes for singular varieties......Page 31
4.4. Obstruction cocycles and classes......Page 34
5.1. Nash transformation......Page 35
6. MacPherson and Mather classes......Page 37
6.2. MacPherson classes......Page 38
7.1. The projective cone......Page 39
7.2. Schwartz-MacPherson classes of the projective cone......Page 41
7.3. Case of the Segre and Veronese embeddings......Page 42
8. Polar varieties......Page 43
9. Chern classes via polar varieties (smooth case)......Page 44
10. Mather classes via polar varieties......Page 46
References......Page 47
CLASS 1: Examples etc......Page 50
CLASS 2: Blowups......Page 53
CLASS 3: Transforms......Page 57
The combinatorial handicap......Page 61
The transversal handicap......Page 67
CLASS 5: Resolution of Mobiles......Page 73
References......Page 79
1. Introduction......Page 88
2.1.1. Stratification of the discriminant......Page 97
2.1.2. Classification of quadrics, cubics definition of local type......Page 98
2.2. Fundamental groups of the complements......Page 99
2.2.2. Examples of calculations of the fundamental groups......Page 100
2.2.3. Alexander invariants of the fundamental groups......Page 101
2.2.4. Alexander polynomials of plane algebraic curves: divisibility theorems......Page 103
2.2.5. Alexander polynomials of plane algebraic curves: position of singularities......Page 105
2.3.1. Commutativity in terms of local type of singularities. Nori’s theorem......Page 109
2.3.2. On a proof of Nori’s theorem......Page 111
2.4.1. Action of the fundamental group on higher homotopy groups......Page 112
2.4.2. Orders of the homotopy groups......Page 113
3.1. Van Kampen theorem and braid monodromy......Page 114
3.2. Homotopy groups via pencils......Page 117
3.3. Variation operators......Page 120
4.1.1. Definitions......Page 122
4.1.2. Unbranched covering......Page 124
4.3. Links of isolated non normal crossings......Page 126
5.1. Zeros of Fitting ideals and Hodge numbers in cyclic case......Page 130
5.2. Theory of adjoints......Page 133
5.3. Ideal of quasiadjunction and log-quasiadjunction......Page 136
5.4. Multiplier ideals and log-canonical thresholds......Page 140
5.5. Hodge decomposition for INNCs......Page 142
6. Homotopy groups of the complements to hypersurfaces in projective space and linear systems determined by singularities......Page 143
6.1. Homotopy groups of the complements to INNC......Page 144
6.2. Jumping loci on quasiprojective varieties......Page 145
6.3. The Hodge numbers of abelian covers of projective spaces and linear Systems......Page 146
6.4. Mixed Hodge structure on homotopy groups......Page 148
6.6. Main theorem and Open Problems......Page 149
References......Page 150
1. Stratifications......Page 156
2. Whitney’s conditions (a) and (b)......Page 157
3.1. Transversal intersection of stratifications......Page 160
4. Lipschitz stratifications......Page 162
5. Definable trivialisations......Page 165
6. Bekka’s (c)-regularity......Page 166
7. Condition (tk)......Page 167
8. Density and normal cones......Page 168
References......Page 170
Lagrangian and Legendrian Singularities V. V. Goryunov and V. M. Zakalyukin......Page 174
1.1. Symplectic geometry......Page 176
1.2. Contact geometry......Page 182
Wave front propagation......Page 186
2.1. Lagrangian case......Page 187
2.2. Legendrian case......Page 190
2.3. Examples of generating families......Page 192
Caustics......Page 194
Wave fronts......Page 195
3.2. Metamorphosis of wave front......Page 197
3.3. Affine generating families......Page 198
References......Page 202
Part II: Applications of Singularity Theory......Page 204
1. Robot Manipulators......Page 206
2. Geometry of Manipulator Kinematics......Page 208
3.1. Serial manipulators......Page 212
3.2. Planar 4-bar mechanisms......Page 213
3.3. The Gough-Stewart Platform......Page 214
4. Instantaneous Kinematics and Singularities......Page 216
5.1. One-genericity......Page 219
5.2. Trajectory singularities and a transversality theorem......Page 220
5.3. Problems with genericity......Page 221
6. Screw Systems......Page 222
7. Instantaneous Singular Sets and Applications......Page 225
8. Conclusion......Page 228
References......Page 229
1. Introduction......Page 236
1.1. Equivariant transversality......Page 237
1.2. Applications to vector fields and bifurcation theory......Page 239
2.1. Smooth G-actions......Page 240
2.2. Equivariant vector fields......Page 241
2.3. Representations......Page 242
2.4. Smooth invariant theory......Page 243
2.5. Semialgebraic sets and stratifications......Page 244
3. Local theory of equivariant transversality......Page 246
3.2. The local definition of G-transversality......Page 249
4. Applications to equivariant bifurcation theory......Page 251
Acknowledgements......Page 255
References......Page 256
1. Introduction......Page 258
2. Hypersurfaces in Euclidean space......Page 259
3. Height functions and distance squared functions......Page 262
4. Evolutes and Cylindrical pedals as Caustics and Wave fronts......Page 265
5. Contact with model hypersurfaces and families of model hypersurfaces......Page 270
6. The theory of contact from the view point of Lagrangian or Legendrian singularity theory......Page 273
7. Surfaces in 3-space......Page 277
Appendix A. The theory of Lagrangian singularities......Page 285
Appendix B. The theory of Legendrian singularities......Page 287
References......Page 291
1. Caustics in Visualization Techniques......Page 294
2. Caustics as Singularities......Page 295
3. Calculating Caustics......Page 296
4. Application to the Visualization of Conective Structure in Nematics......Page 298
References......Page 300
1. Projection diagnostic radiology and signs for interpretation......Page 302
1.1. The silhouette sign for chest roentgenology......Page 304
1.2. Justifying signs: singularity theory and genericity......Page 305
1.3. The three stable signs: applications......Page 306
2. Extending control......Page 307
2.1. The formalism of controlled mappings......Page 308
2.2. Controlled projections: generic results......Page 309
3.1. Controlled sections and stacks of parallel sections......Page 310
4.2. Shape coding......Page 311
5.1. Adaptive branching in nature, generalized catastrophes, simple models......Page 312
6.1. Algorithm......Page 313
6.3. Abortive bifurcations......Page 314
7.1. Algorithm......Page 316
7.2. Experimental results......Page 317
8.1. Application: curve to surface matching and Interventional Imaging......Page 318
9. Conclusions and Prospects......Page 320
Reference......Page 321
1. Introduction......Page 324
2.1. Second fundamental form of surfaces in Rn......Page 325
2.2. The curvature ellipse......Page 326
2.3. The rank of the second fundamental form of surfaces in R5......Page 327
3. 2-singular points......Page 329
4. Extrinsic dynamics: v-principal configurations......Page 330
5. Contacts with hyperplanes......Page 332
6. Contact directions......Page 333
7. Essential convexity......Page 335
8. Contacts with hyperspheres......Page 336
9. 2-regularity for surfaces in S4......Page 338
10. Isometric reduction of the codimension and 2-regularity of surfaces in Rn......Page 339
References......Page 340
1.1. Various contexts......Page 344
1.2. Methodology: generic versus concrete systems......Page 347
1.3. Related work......Page 348
2.1. Background and sketch of results......Page 349
2.2. Reduction to an equivariant bifurcation problem......Page 350
2.3. Zq singularity theory......Page 352
2.4. Resonance domains......Page 354
3.1. A Normal Form Algorithm......Page 356
3.2. Applications of the Normal Form Algorithm......Page 359
3.3. Via covering spaces to the Takens Normal Form......Page 365
4. Generic Hopf-Neimark-Sacker bifurcations in feed forward systems?......Page 367
5. Conclusion and future work......Page 370
References......Page 371
1.1. Motivations for the setting of the problem......Page 374
1.3. The notion of a \"general position”......Page 376
1.5. How to use our approach in applications?......Page 378
2. Singularities of a generic product function......Page 379
3. Stable singularities of surfaces......Page 383
4. Near-singularities and Organizing Center......Page 387
References......Page 392
Part III: Geometry and Topology of Singularities......Page 394
1. Introduction......Page 396
Combinatorial Hard Lefschetz Theorem (HLT)......Page 397
Hodge-Riemann bilinear relations (HRR)......Page 398
2. Outline of the proof of the Hodge-Riemann relations......Page 399
3. Cutting off......Page 400
4. Intersection Cohomology of Fans......Page 405
5.1. HRR for pyramids......Page 410
5.2. The Kunneth formula......Page 412
5.3. Transversal Cuttings......Page 415
6. Deformation......Page 419
References......Page 426
On Rational Cuspidal Plane Curves, Open Surfaces and Local Singularities J. Fernandez de Bobadilla, I. Luengo, A. Melle-Hernandez and A. Nemethi......Page 428
1. Introduction......Page 429
2.1.2. Equisingular deformations......Page 431
2.2. Local invariants and embedded resolution......Page 432
3.1. Expected dimension......Page 433
3.2. The action of PGL(3,C)......Page 434
3.3. The Coolidge-Nagata conjecture......Page 435
4. On rational plane curves......Page 436
5. Cuspidal rational plane curves and the Rigidity Conjecture......Page 438
5.2. Rational unicuspidal plane curves with one Puiseux pair......Page 439
5.4. The rigidity conjecture of Flenner and Zaidenberg......Page 440
6.1. Compatibility property......Page 443
6.3. A counterexample to an ‘extended’ version......Page 445
7.1. Superisolated singularities......Page 447
7.2. Normal surfaces whose link is a rational homology sphere......Page 448
7.3. Seiberg- Witten invariant of a superisolated singularity......Page 450
8.1. Graded roots......Page 451
8.3. The canonical graded root (R, X) of M. [32]......Page 452
8.5. Example......Page 453
8.6. The Z[U]-module associated with a graded root......Page 454
Acknowledgments......Page 455
References......Page 456
1. Introduction......Page 460
2.1.1. Zero Angular Momentum form particles in R2......Page 463
2.2. Hamiltonian actions of nonabelian Lie groups......Page 464
2.2.1. Commuting Varieties......Page 465
3. Koszul resolution......Page 466
5. The quantum BRST-algebra......Page 471
6. Quantum reduction......Page 472
Appendix A. Two perturbation lemmata......Page 475
Bibliography......Page 477
1. Introduction......Page 480
2. Differentiability and blow-up......Page 482
3.1. General bounds for a class of quasihomogeneous maps......Page 487
3.2. Some special homogeneous maps. The Newton map......Page 488
3.3. Holomorphic mappings of the plane......Page 489
References......Page 490
1. Introduction......Page 492
2.1. Definitions......Page 493
2.2. Singular vs. smooth Milnor fillability......Page 494
2.3. Smooth Milnor fillability vs. Stein fillability......Page 495
3.2. Non unicity......Page 496
3.3. Milnor open books on smoothly Milnor fillable contact 3-manifolds......Page 498
Acknowledgements......Page 500
References......Page 501
1. Introduction......Page 504
2. The normal form and realization theorems......Page 506
3. The canonical orientation and the 2-dimensional kernel of w at 0......Page 512
4. Determination by the restriction of w to T   and the canonical orientation......Page 513
Acknowledgements......Page 517
References......Page 518
Introduction......Page 520
1. La filtration par le poids et les halos......Page 522
1.1. Decomposition de certains Z [t, t-1] -modules......Page 523
1.2. Invariant complet de la structure isometrique associee a un halo de valence 3 et unites cyclotomiques......Page 528
1.3. Determination d’un halo de valence 3......Page 529
2. L\'arbre de desingularisation au voisinage du sommet #1......Page 531
3. Reconstruction de l’arbre reduit de l’arbre T(f)......Page 544
4. Calcul des paires de Zariski et fin de la reconstruction de I’arbre de desingularisation......Page 551
5.1. Lemmes preliminaires......Page 558
5.2. Matrice de presentation de M-zH1(F,Z)......Page 562
5.3. Gerrnes isomhres......Page 563
6.1. Les donnees......Page 567
6.2. Calcul des unites......Page 568
6.3. Les solutions du probltme......Page 570
References......Page 571
Introduction......Page 574
1. Special points of 1-forms......Page 575
2. Local Chern obstructions......Page 577
References......Page 580
1. Introduction......Page 582
2. Preliminaries......Page 585
3. Proofs of Theorem 1.1 and Lemma 1.1......Page 586
References......Page 592
Introduction......Page 594
1. Resolution of Singularities and the Topological   -Function......Page 595
Identification of Exceptional Divisors......Page 600
Euler characteristic......Page 603
References......Page 607
1.1. Implicit Hamiltonian systems......Page 610
1.2. Main theorems......Page 611
2.1. Integrability condition in terms of a generating family......Page 613
2.2. Normal forms of fold singularities......Page 616
3.2. Proof of Theorem 1.2......Page 618
4.1. A method for constructing smoothly integrable global Lagrangian immersions......Page 619
4.2. Example: A compact orientable surface with genus two......Page 621
4.3. Example: An integrable Lagrangian immersion of a sphere......Page 624
References......Page 628
1.1. Arc spaces......Page 630
1.2. Grothendieclc rings......Page 631
2. Motivation......Page 633
3. Motivic vanishing cycles morphism on the Grothendieck group......Page 634
3.1. A modified zeta function......Page 635
4.1. Iterated vanishing cycles......Page 636
6. Spectrum and the Steenbrink conjecture......Page 637
7. A computation: Motivic Milnor fiber of a non-degenerate composite......Page 638
Reference......Page 639
1. Introduction......Page 642
2. On local triviality of gl(X \\ Y)......Page 645
3. Transition from glX \\ Y to glX......Page 652
4. Local triviality of glX in the case dimX = 2, Y smooth......Page 658
References......Page 665
1. Introduction......Page 668
2.1. Algebra......Page 670
2.2. Subgroups of Sk......Page 672
2.3. Topology......Page 673
3. Multiple Point Spaces of Finite Maps......Page 677
4.1. Semi-simplicial resolution......Page 680
4.3. Homology of the filtration......Page 684
5. Spectral Sequences......Page 687
References......Page 691
1. Polar varieties and Limits of Tangents......Page 694
2. Constructing generic hyperplanes......Page 696
References......Page 699
1. Introduction......Page 700
2. Preliminaries......Page 703
3. Radial vector fields......Page 706
4. Local model......Page 708
5. Stable/unstable manifold......Page 711
6. Stratified gradient-like vector fields......Page 713
7. The space of trajectories between singular points......Page 719
8. Morse-Smale-Witten complex for a stratified gradient-like vector field......Page 725
References......Page 729
1. Introduction......Page 732
2. Examples of obstructed families......Page 733
3. Some remarks on conditions for T-smoothness......Page 735
4. Some remarks on cones......Page 737
References......Page 740
1. Introduction......Page 742
2.1. Preliminaries......Page 744
2.2. Intersection homology approach......Page 746
2.3. The Milnor fiber of a projective hypersurface arrangement......Page 751
3. Universal abelian Alexander invariants of the complement......Page 752
3.1. Definition of Characteristic varieties......Page 753
3.2. Further study of Supports......Page 755
3.3. Dependence on the local data......Page 757
References......Page 758
1. Introduction......Page 762
2. Plumbing graphs......Page 764
3. Vertical monodromies......Page 765
4. The Seifert structure on the boundary of the Milnor fiber......Page 769
5. Examples......Page 776
References......Page 777
1. Introduction......Page 778
3. Transversality for Whitney stratifications......Page 780
4. Further geometric applications of the Goresky Theorem......Page 785
5. Improvement of the Goresky Theorem. Applications. Open problems......Page 788
5.1. Preservation of regularity after deformation Open problems......Page 794
5.2. Stratification of the transverse union and intersection. Open problem......Page 795
6. Transverse intersections and other applications......Page 796
6.1. Application to abstract stratified and (c)-regular homology......Page 797
6.2. Some applications to homotopy of stratified spaces......Page 798
7. More on Goresky’s stratified transversality. Supertransversality......Page 799
References......Page 801
1. Introduction......Page 804
2. The place of graph manifolds in 3-manifold theory......Page 805
3. Seifert manifolds......Page 806
4. Decomposition graphs, decomposition matrices......Page 808
5. Splice diagrams for rational homology spheres......Page 811
6. Singularities of splice type......Page 813
7. Some basics......Page 818
8. JSJ Decomposition......Page 821
9.1. Orbifolds......Page 827
9.3. Seifert circle fibrations......Page 828
9.4. “Seifert fibmtions” with torus fiber......Page 830
9.5. Simple Seifert fibered manifolds......Page 831
References......Page 832
1. Introduction......Page 836
2.1. Igusa\'s zeta function......Page 837
2.3. The Monodromy Conjecture......Page 838
3.1. The Grothendieck ring of varieties......Page 839
3.2. Arc spaces......Page 840
3.3. Motivic zeta functions......Page 841
3.4. The motivic Monodmmy Conjecture......Page 842
4.1. Construction of the analytic Milnor fiber......Page 843
4.2. Points of the analytic Milnor fiber......Page 844
4.3. Cohomology of the analytic Milnor fiber......Page 845
5.1. Counting points on rigid spaces: the motivic Serre invariant......Page 846
5.2. A Trace formula......Page 847
5.3. Computation of the Serre Poincare series......Page 848
5.4. Motivic Serre invariant over the separable closure......Page 849
5.5. Applications to the motivic zeta function......Page 850
Bibliography......Page 852
1. Introduction......Page 854
2. Reducible sextics of non-torus type......Page 856
3. Configuration coming from quintic flex geometry......Page 858
3.1. Example for sextics with quintic components......Page 860
4.1. Configuration coming from quartic flex geometry......Page 865
4.2. Conical geometry of quartic......Page 867
5.1. Configurations coming from cubic flex geometry: a cubic component and a line component......Page 871
5.2. Sextics with two cubic components: C = B3+B......Page 874
5.4. Examples of (b) and (d)......Page 877
6. Three conics......Page 878
References......Page 880
1. Introduction......Page 882
2. Euler-Poincare characteristic......Page 884
3. Characteristic classes of vector bundles......Page 888
4. Characteristic classes of smooth manifolds......Page 892
5. Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch......Page 895
6. The Generalized Hirzebruch-Riemann-Roch......Page 898
7. Characteristic classes of singular varieties......Page 901
7.1. Stiefel-Whitney classes w*......Page 902
7.2. Chern classes c*......Page 903
7.3. Todd classes td*......Page 905
7.4. L-classes L*......Page 906
8. Relative Grothendieck rings of varieties and motivic characteristic classes......Page 911
9. Bivariant Characteristic classes......Page 918
10. Characteristic classes of proalgebraic varieties......Page 933
11. Stringy and arc characteristic classes of singular spaces......Page 943
11.1. Elliptic classes......Page 947
11.2. Motivic integration......Page 952
11.3. Stringy/arc E-function and Euler characteristic......Page 956
11.4. Stringy and arc characteristic classes......Page 959
Acknowledgments......Page 962
References......Page 963
1. Introduction......Page 970
2. The Schwartz index......Page 973
3. The local Euler obstruction......Page 977
4. The GSV-index......Page 980
5. The Virtual Index......Page 982
6. The Homological Index......Page 984
7. Relations with Chern classes of singular varieties......Page 987
References......Page 990
1. Introduction......Page 994
2.1. The Cube Construction......Page 995
3. Recall of Periodic Cyclic Homology......Page 999
4. Distance Function and Chern Character......Page 1001
4.1. Riemannian Geometry Preliminaries......Page 1002
4.2. The Characteristic Cyclic Functions  k......Page 1003
4.3. The Characteristic Cyclic Functions  k and Chern Character......Page 1004
5. Direct Connections and Chern Character.......Page 1008
References......Page 1010
Introduction......Page 1012
1.1. The case of weighted homogeneous hypersurfaces with an isolated singularity......Page 1013
1.3. The case of free divisors......Page 1014
2.1. Preliminaries......Page 1016
2.2. On the perversity of s1>(log 0)......Page 1017
2.3. A differential characterization of LCT(D)......Page 1018
3.1. Preliminaries......Page 1019
3.2. The conditions B(hD) and LCT(D)......Page 1020
3.3. The conditions A(l/hD) and LCT(D)......Page 1021
3.4. The condition A(l/h)......Page 1022
Appendix A......Page 1023
References......Page 1024
1. Introduction......Page 1028
3. Frobenius endomorphism......Page 1031
4. Torsion......Page 1033
5. Weight filtration and gradation......Page 1034
References......Page 1035
1. The centre symmetry sets......Page 1036
2. Focal loci......Page 1038
3. Algebraic approach to centre symmetry sets......Page 1039
3.1. ACSS of an algebraic curve – cubic case......Page 1040
3.2. ACSS of an algebraic curve – another approach......Page 1044
3.4. CSS of an algebraic curve – preliminary remarks......Page 1045
References......Page 1047
Pictures of the Participants......Page 17
Programs......Page 1058
List of Participants......Page 1068