Singularities and Topology of Hypersurfaces

ISBN-13: 9780387977096 / Angielski / Miękka / 1992 / 263 str.

Alexandru Dimca

Opis

From the very beginning, algebraic topology has developed under the influ ence of the problems posed by trying to understand the topological properties of complex algebraic varieties (e.g., the pioneering work by Poincare and Lefschetz). Especially in the work of Lefschetz Lf2], the idea is made explicit that singularities are important in the study of the topology even in the case of smooth varieties. What is known nowadays about the topology of smooth and singular vari eties is quite impressive. The many existing results may be roughly divided into two classes as follows: (i) very general results or theories, like stratified Morse theory and (mixed) Hodge theory, see, for instance, Goresky-MacPherson GM], Deligne Del], and Steenbrink S6]; and (ii) specific topics of great subtlety and beauty, like the study of the funda mental group of the complement in p>2 of a singular plane curve initiated by Zariski or Griffiths' theory relating the rational differential forms to the Hodge filtration on the middle cohomology group of a smooth projec tive hypersurface. The aim of this book is precisely to introduce the reader to some topics in this latter class. Most of the results to be discussed, as well as the related notions, are at least two decades old, and specialists use them intensively and freely in their work. Nevertheless, it is impossible to find an adequate intro duction to this subject, which gives a good feeling for its relations with other parts of algebraic geometry and topology.

Spis treści

1 Whitney Stratifications.- 1. Some Motivations and Basic Definitions.- 2. Topological Triviality and ?*-Constant Deformations.- 3. The First Thom Isotopy Lemma.- 4. On the Topology of Affine Hypersurfaces.- 5. Links and Conic Structures.- 6. On Zariski Theorems of Lefschetz Type.- 2 Links of Curve and Surface Singularities.- 1. A Quick Trip into Classical Knot Theory.- 2. Links of Plane Curve Singularities.- 3. Links of Surface Singularities.- 4. Special Classes of Surface Singularities.- 3 The Milnor Fibration and the Milnor Lattice.- 1. The Milnor Fibration.- 2. The Connectivity of the Link, of the Milnor Fiber, and of Its Boundary.- 3. Vanishing Cycles and the Intersection Form.- 4. Homology Spheres, Exotic Spheres, and the Casson Invariant.- 4 Fundamental Groups of Hypersurface Complements.- 1. Some General Results.- 2. Presentations of Groups and Monodromy Relations.- 3. The van Kampen-Zariski Theorem.- 4. Two Classical Examples.- 5 Projective Complete Intersections.- 1. Topology of the Projective Space Pn.- 2. Topology of Complete Intersections.- 3. Smooth Complete Intersections.- 4. Complete Intersections with Isolated Singularities.- 6 de Rham Cohomology of Hypersurface Complements.- 1. Differential Forms on Hypersurface Complements.- 2. Spectral Sequences and Koszul Complexes.- 3. Singularities with a One-Dimensional Critical Locus.- 4. Alexander Polynomials and Defects of Linear Systems.- Appendix A Integral Bilinear Forms and Dynkin Diagrams.- Appendix B Weighted Projective Varieties.- Appendix C Mixed Hodge Structures.- References.