This book explores the topological properties of algebraic varieties, focusing on both smooth and singular varieties. It introduces key topics in algebraic topology, emphasizing their connections to other areas of algebraic geometry. The content is aimed at providing a comprehensive introduction to these intricate subjects.
This textbook provides an accessible introduction to the rich and beautiful area of hyperplane arrangement theory, where discrete mathematics, in the form of combinatorics and arithmetic, meets continuous mathematics, in the form of the topology and Hodge theory of complex algebraic varieties. The topics discussed in this book range from elementary combinatorics and discrete geometry to more advanced material on mixed Hodge structures, logarithmic connections and Milnor fibrations. The author covers a lot of ground in a relatively short amount of space, with a focus on defining concepts carefully and giving proofs of theorems in detail where needed. Including a number of surprising results and tantalizing open problems, this timely book also serves to acquaint the reader with the rapidly expanding literature on the subject. Hyperplane Arrangements will be particularly useful to graduate students and researchers who are interested in algebraic geometry or algebraic topology. The book contains numerous exercises at the end of each chapter, making it suitable for courses as well as self-study.
Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties). This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant)coefficients. The first 5 chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. Later chapters apply this powerful tool to the study of the topology of singularities, polynomial functions and hyperplane arrangements. Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the basic theory to current research questions, supported in this by examples and exercises.
Inhaltsverzeichnis1: Two Classical Examples: Submersion Theorem and Morse Lemma.2: Germs and Jets.3: Equivalence Relations on Germs and Jets.4: Tangent Spaces to Orbits.5: Basic Classification Examples in Linear Algebra and Algebraic Geometry.6: Finite Determinacy of Map Germs.7: Weighted Homogeneous Singularities.8: Classification of Simple Singularities of Functions.9: Classification of Simple O-Dimensional Complete Intersections.10: Curve and Surface Singularities.11: Dual Mappings and Contact Tangency Classes for Projective Hypersurfaces.Appendix: Two Recent Research Topics.References.List of Notations.