Singularities in geometry and topology : proceedings of the Trieste Singularity Summer School and Workshop, ICTP, Trieste, Italy, 15 August - 3 September 2005

Singularity theory appears in numerous branches of mathematics, as well as in many emerging areas such as robotics, control theory, imaging, and various evolving areas in physics. The purpose of this proceedings volume is to cover recent developments in singularity theory and to introduce young researchers from developing countries to singularities in geometry and topology. The contributions discuss singularities in both complex and real geometry. As such, they provide a natural continuation of the previous school on singularities held at ICTP (1991), which is recognized as having had a major influence in the field. CONTENTS......Page 10 Introduction......Page 6 Part I Elementary School on Singularity Theory......Page 14 Introduction......Page 16 1.1. Group embeddings......Page 17 1.2. Toric varieties: Basic examples......Page 20 1.3. Characters and one-parameter subgroups of tori......Page 23 2.1. Algebraic description: The coordinate ring......Page 27 2.2. Geometric description: Polyhedral lattice cones......Page 30 2.3. Cones and orbit structure......Page 39 3.1. Local structure at a fixed point......Page 47 3.2. General toric varieties and fans......Page 53 3.3. Resolution of toric singularities......Page 60 References......Page 69 1. Introduction......Page 70 2.1. Combinatorial definition......Page 71 2.1.1. Betti numbers......Page 73 3.1. The index: index as a degree......Page 75 3.1.2. Vector fields on a manifold......Page 76 3.2. The index - Definition by obstruction theory......Page 77 3.3.1. In Euclidean space......Page 78 3.3.2. In a manifold......Page 79 3.4. Relation with the Gauss map......Page 80 4.1. The smooth case without boundary......Page 82 4.1.1. Consequences of Poincare-Hopf Theorem......Page 83 4.2. The smooth case with boundary......Page 84 5.1. Poincare-Hopf Theorem fails in general......Page 85 5.2. Whitney stratifications......Page 86 5.3. Radial extension process - the local case......Page 87 5.4. Poincard-Hopf Theorem for singular varieties......Page 89 References......Page 92 1. Manifolds and Local, Ambient, Topological-type......Page 94 2. The Geometry of the Implicit Function Theorem......Page 97 3. The Theorem of Ehresmann and Integrating Vector Fields......Page 100 4. Basic Morse Theory......Page 102 5. Real and Complex Analytic Sets......Page 108 6. Real and Complex Semianalytic Sets......Page 124 7. Partitions and Stratifications......Page 127 8. Basic Stratified Morse Theory......Page 134 References......Page 138 On Milnor’s Fibration Theorem for Real and Complex Singularities J. Seade......Page 140 1. An example: Pham-Brieskorn singularities......Page 143 2. The classical fibration theorem of Milnor......Page 148 3. A glance at generalisations......Page 152 4. Real analytic germs with a Milnor fibration......Page 153 5. Twisted Pham-Brieskorn singularities......Page 156 6. A method for studying maps into R2......Page 160 7. A refinement of Milnor's classical theorem......Page 163 8. Singularities fg and Milnor fibrations for meromorphic germs......Page 165 9. Fibrations of multilinks......Page 167 References......Page 169 Introduction to Complex Analytic Geometry T. Suwa......Page 174 1. Analytic functions of one complex variable......Page 175 2. Analytic functions of several complex variables......Page 176 3. Germs of holomorphic functions......Page 180 4. Complex manifolds and analytic varieties......Page 182 5. Germs of varieties......Page 188 1. Vector bundles......Page 191 2. Vector fields and differential forms......Page 197 4. de Rham cohomology......Page 200 5. Cech-de Rham cohomology......Page 202 1. Chern classes via connections......Page 207 2. Virtual bundles......Page 210 3. Characteristic classes in the Cech-de Rham cohomology and a vanishing theorem......Page 211 4. Divisors......Page 212 5. Complete intersections and local complete intersections......Page 213 6. Grothendieck residues......Page 216 1. Localization of the top Chern class......Page 217 (I) Analytic expression......Page 219 (III) Topological expression......Page 220 (b) Multiplicity formula......Page 221 4. Residues of Chern classes on singular varieties......Page 222 5. Residues at an isolated singularity......Page 224 (I) Analytic expression......Page 225 (III) Topological expression......Page 226 (a) Index of a holomorphic 1-form of Ebeling and Gusein-Zade......Page 227 (b) Multiplicity of a function on a local complete intersection......Page 228 (c) Some others......Page 230 References......Page 231 Part II Advanced School on Singularity Theory......Page 234 1. Metric viewpoint. Comparison of metrics. Normal embedding.......Page 236 2. Finiteness results.......Page 238 3. Germs of subanalytic surfaces.......Page 239 4. Metric homology.......Page 242 5. Characteristic exponents of germs of subanalytic sets.......Page 243 Acknowledgments......Page 244 References......Page 245 Introduction......Page 247 1. Geometric Monodromy......Page 248 2. Deformation and monodromy......Page 250 3. Resolution and monodromy......Page 258 4. C*-action and monodromy......Page 262 References......Page 264 Preface......Page 266 1. Studying the Singular Locus......Page 267 1.1. The Jacobian Criterion......Page 268 1.2. The Non-equidimensional Case......Page 274 1.3. Finding the Comct Number of Components......Page 277 2.1. Local and Global Considemtions......Page 281 2.2. Dimension and Multiplicity......Page 284 2.3. Milnor and Tjurina Number......Page 286 2.4. Pzliseua: Ezpansion......Page 289 2.5. Classification of Hgpersurface Singularities......Page 293 2.6. Monodmmy and Spectral Numbers......Page 294 3.1. T1 andT2......Page 297 3.2. Studying Families of Singularities......Page 305 4. Varieties with Singularities......Page 310 4.1. Hypersurfaces with Prescribed Singularities......Page 311 4.2. Resolution of Singularities......Page 313 4.2.1. Theoretical Background......Page 314 4.2.2. Blowing Up......Page 320 4.2.3. Computing the Locus of Maximal Order......Page 324 4.2.4. Descent in Dimension......Page 327 4.2.5. Identification of Exceptional Divisors......Page 329 4.2.6. Intersection Matrix of Exceptional Curves......Page 331 References......Page 338 Lagrangian and Legendrian Varieties and Stability of Their Projections V. V. Goryunov and V. M. Zakalyukin......Page 341 1.1. Symplectic geometry......Page 343 1.2. Contact geometry......Page 349 2.1. Lagmngian case......Page 352 3.1.1. The Lagrangian setup......Page 355 3.1.2. The Legendrian case......Page 359 3.2.1. The critical-value theorem......Page 360 3.2.2. Composite functions......Page 363 References......Page 365 1. Introduction......Page 367 2. A resolution of the singularities of a toric variety......Page 368 3. A weak resolution of a non-degenerate hypersurface......Page 370 4. Canonical divisors......Page 373 5. Singularities on a non-degenerate hypersurface......Page 377 References......Page 381 1. Introduction......Page 383 2. Alexander Invariants......Page 384 2.1. Realization PTO blems......Page 385 3. Fundamental Groups......Page 386 3.3. Braid monodromy......Page 387 3.4. of the complements to generic projections......Page 390 3.6. Question of Artin and Mazur......Page 392 4. Multivariable Alexander Invariants......Page 393 5.1. Dimensions of components of characteristic varieties......Page 395 5 . 2 . Arrangement Strata......Page 396 References......Page 398 1. Degeneration of Riemann surfaces and topological monodromy......Page 401 2. Generalized quotients......Page 404 References......Page 405 1. Introduction.......Page 407 2.1. The link.......Page 409 2.2. The combinatorics of the link.......Page 411 2.3. The topology of the link. The Heegaard Floer homology HF+(--M).......Page 414 2.4. Some analytic invariants of the singularity.......Page 416 2.5. Rational singularities.......Page 418 2.6. Weakly elliptic singvlarities.......Page 419 2.7. Almost rational singularities.......Page 421 3.1. Graded roots.......Page 422 3.2. The homology of a graded root.......Page 424 3.3. Graded roots associated with plumbing graphs.......Page 426 3.4. Characterization of rational and elliptic graphs via roots; classification.......Page 427 3.5. Graded roots and Heegaard-Floer homology of almost rational graphs.......Page 430 3.6. Example. Lens spaces.......Page 431 4.1. Introduction.......Page 434 4.2. Line bundles on X.......Page 435 4.3. Some cohomological computations.......Page 439 4.4. Main (conjectured) properties.......Page 441 4.5. Example. The case of rational singularities.......Page 445 5.2. The manifold S3_p/q(K f ) .......Page 447 5.3. The main invariants of S3_p/q (Kf)......Page 451 5.4. The first part of the proof of 5.3.2: k2r + #j.......Page 453 5.5. The second part of the proof of 5.3.2: (R [k], X [h]).......Page 458 5.6. Examples.......Page 463 5.7. S_p/q(K) as Kulikov graph-manifold.......Page 465 6.1. The semigroup distribution property.......Page 467 6.2. The semigroup distribution property and surface singularities.......Page 469 6.3. The semigroup distribution property and graded roots/Heegaard Floer homology.......Page 470 References......Page 473 1. Introduction......Page 477 2.1. Chern classes of vector bundles......Page 479 2.2. Chern class for singular varieties......Page 480 3.1. Totaro's construction of BG......Page 481 3.2. G -equivariant (co)homology......Page 482 3.3. Equivariant constructible functions......Page 483 3.4. Equivariant natrual transformation......Page 484 4.1. Universality......Page 485 4.2. Segre-Schwartz-MacPherson class......Page 486 4.3. Singularities of maps......Page 487 5.1. Canonical constructible functions......Page 489 5.3. Partitions......Page 490 5.5. Exponential formula......Page 491 5.6. Examples......Page 492 References......Page 493 1. Platonic triples and simple objects......Page 496 2. Klein singularities......Page 498 3. McKay’s construction......Page 499 4. The phenomenological McKay correspondence......Page 502 5. The geometric McKay correspondence for quotient surface singularities......Page 514 6. The construction of Ito and Nakamura......Page 522 7. Cyclic quotient surface singularities......Page 524 8. Interpretation in terms of derived categories and A. Ishii’s result......Page 526 9. What remains to do?......Page 529 Bibliography......Page 531 1. Introduction......Page 533 2.1. Singularities at infinity......Page 535 2.2. A bouquet theorem......Page 538 3. Regularity conditions at infinity......Page 541 3.1. p-regularity and t-regularity......Page 542 3.2. The relation to Malgrange condition......Page 547 4. Polar curves and regularity conditions......Page 548 5.1. The complex setting......Page 551 5.2. The real setting......Page 552 6. Families of complex polynomials with singularities at infinity......Page 557 7. Singularity exchange at infinity......Page 558 7.1.2. Semi-continuity at infinity......Page 559 7.2. Persistence of -singularities......Page 561 7.3.1. Rigidity in deformations with constant +......Page 562 7.4.1. F-class examples; behaviour of......Page 563 7.4.2. B-class examples......Page 564 References......Page 565 1. Introduction......Page 569 2. Weights in t-adic cohomology......Page 570 3. Mixed Hodge theory......Page 571 4. From groups to sheaves......Page 577 5. Intersection homology and perverse sheaves......Page 580 6. D-modules......Page 582 6.1. Good Filtrations and Characteristic Varieties......Page 584 6.2. Basics on Holonomic D-Modules......Page 586 6.3. De Rham finctor and Riemann-Hilbert correspondence......Page 587 7.1. Motivating example......Page 588 7.2. Axioms for mixed Hodge modules......Page 589 7.3. Some Consequences of the Axioms......Page 591 8. Some Applications......Page 592 References......Page 593 Part III Workshop on Singularities in Geometry and Topology......Page 596 1. Introduction......Page 598 2. Right left-sufficiency of jets from the plane to the plane......Page 602 3. Sufficiency of jets with line singularities......Page 607 References......Page 609 1. Introduction......Page 611 2.1. Stationary strategies......Page 613 2.2. Cyclic motion......Page 617 3. Averaged Profit singularities for level cycles......Page 619 3.1. Monotonicity and continuity of the profit and the period for level cycles......Page 620 3.2. Diflerentiability of the averaged profit for level cycles......Page 621 3.3. Singularities of the averaged profit for optimal level cycles......Page 623 3.4. Transition through a swallow point......Page 627 4. Singularities of stationary strategies......Page 631 5.1. Transition between optimal strategies......Page 635 Acknowledgments......Page 639 References......Page 640 1. Introduction......Page 642 2. The GSV index......Page 644 3. Local Chern obstructions......Page 647 References......Page 651 Introduction......Page 653 1. The main result......Page 654 2. Analytic parafactoriality......Page 656 3. Reduction to the smooth quasi-projective case......Page 665 4. Use of the exponential sequence......Page 667 Bibliography:......Page 671 singularity theory......Page 674 ADE-case......Page 676 Thorn-Sebastiani property......Page 679 Bibliography......Page 680 1. Introduction......Page 682 2.1. First homology group H1(P2 - C ) .......Page 683 2.3. Product formula......Page 684 2.5. Class formula and flex formula......Page 685 3.2. Alexander polynomials of plane curves......Page 686 3.3. Fox calculus......Page 687 3.4. Weakness of the generic Alexander polynomial c(t)......Page 688 4.1. Dual stratification of curves......Page 689 4.2. Dual stratification of the configuration space ( , d )......Page 690 4.5. Example......Page 691 4.6. -Alexander polynomials......Page 694 4.7. Relations between the tangential Alexander polynomials and -Alexander polynomials......Page 696 5 .I. Admissible polydisk......Page 698 5.2. Two non-reduced degenerations......Page 699 5.3. Surjectivity Theorem for line degenerations......Page 700 5.3.2. Line degenerations of order 2......Page 702 5.4. Line degeneration of curves of torus type......Page 704 5.4.1. Singularities of line-degenerated torus curves......Page 705 5.5.2. Quartic......Page 706 5.5.5. A quartic with A5......Page 707 5.5.8. Quintics as line-degenerations......Page 708 5.6. Sextics as line degenerations......Page 709 5.7. Flex degenerations......Page 710 5.8. Appendix......Page 712 5.9- Graphs of various quartics......Page 713 References......Page 715 1. Introduction.......Page 718 2. Basic notation and statements.......Page 720 3.1. Blow-up of (C2, ( 0 , 0 ) )......Page 723 3.2. Blow-up of germs of vector fields in Vn+1......Page 724 3.3. Properties of the blow-up of dicritic germs of vector fields in Vdn+1.......Page 726 4.1. Tangency points on L.......Page 727 5. Preliminary normalization for generic dicritic germs of vector fields in Vn+l d,gen.......Page 728 6.2. Involutions.......Page 729 6.3. Local first integral.......Page 730 6.4. Local normalizations and end of the proof.......Page 731 References......Page 733 Polar Multiplicities and Euler Obstruction of the Stable Types in Weighted Homogeneous Map Germs from Cn to C3, n 3 E. C. Rizziolli and M. J. Saia......Page 736 1.1. The Stable types......Page 737 1.2. Polar multiplicities......Page 740 2. Weighted homogeneous map germs......Page 749 3. Polar multiplicities in ( f )......Page 750 4. Polar multiplicities in f( (n-2,1)(f))......Page 754 5. The local Euler obstruction of the stable types......Page 757 6. Euler obstruction of simple germs f : (C3,0) (C3,0)......Page 759 References......Page 760 1. Introduction......Page 762 2. Complete intersection of projection......Page 763 3. Multiplication table and the logarithmic vector fields......Page 770 4. Multiplication table and the topology of real hypersurfaces......Page 775 5. Topology of real complete intersections......Page 778 6. Examples......Page 783 References......Page 790 1. Introduction......Page 792 2. Simple and simple elliptic singularities......Page 793 3. Semi-universal deformations via Lie algebras......Page 795 References......Page 799 1. Introduction......Page 800 2. The energy functional on the space of paths......Page 801 3. The cut-locus......Page 803 4. The geodesic flow......Page 804 5. Covariant differentiation and curvature......Page 807 6. Jacobi fields......Page 809 7. Conjugate locus and Jacobi fields......Page 813 9. Known theorems about cut and conjugate loci......Page 814 10. A conjecture of Arnol'd......Page 816 11. The examples of Markatis......Page 817 12. Questions asked by Thorn......Page 819 13. Answers given by Looijenga......Page 821 14. The caustic......Page 823 15. Singularities of the medial axis......Page 824 16. Two lesser known theorems in Thom36......Page 827 17. Singularities of the symmetry set......Page 830 18. A weighted symmetry set......Page 832 References......Page 835 1. What is a curve?......Page 838 2. What does one do with curves?......Page 841 3. Newton’s study of plane curve singularities......Page 844 4. Puiseux exponents......Page 854 5.1. Fitting ideals......Page 858 5.2. A proof of Bekout's theorem (after [T5], $1)......Page 862 6. Resolution of plane curves......Page 866 7.1. Integral dependance......Page 876 7.2. The invariant of a plane curve singularity......Page 878 7.3. projections of space curves......Page 881 8. The semigroup of a branch......Page 883 9. Resolution of binomials......Page 888 10. Relation with topology......Page 892 11. Duality......Page 894 12. The polar curve......Page 897 Bibliography......Page 898 Programme of the Conference......Page 902 List of Participants......Page 908