Complex Analytic Geometry is a subject that could be termed, in short, as the study of the sets of common zeros of complex analytic functions. It has a long history and is closely related to many other fields of Mathematics and Sciences, where numerous applications have been found, including a recent one in the Sato hyperfunction theory.
This book is concerned with, among others, local invariants that arise naturally in Complex Analytic Geometry and their relations with global invariants of the manifold or variety. The idea is to look at them as residues associated with the localization of some characteristic classes. Two approaches are taken for this — topological and differential geometric — and the combination of the two brings out further fruitful results. For this, on one hand, we present detailed description of the Alexander duality in combinatorial topology. On the other hand, we give a thorough presentation of the Čech–de Rham cohomology and integration theory on it. This viewpoint provides us with the way for clearer and more precise presentations of the central concepts as well as fundamental and important results that have been treated only globally so far. It also brings new perspectives into the subject and leads to further results and applications.
The book starts off with basic material and continues by introducing characteristic classes via both the obstruction theory and the Chern–Weil theory, explaining the idea of localization of characteristic classes and presenting the aforementioned invariants and relations in a unified way from this perspective. Various related topics are also discussed. The expositions are carried out in a self-containing manner and includes recent developments. The profound consequences of this subject will make the book useful for students and researchers in fields as diverse as Algebraic Geometry, Complex Analytic Geometry, Differential Geometry, Topology, Singularity Theory, Complex Dynamical Systems, Algebraic Analysis and Mathematical Physics.
Contents:
- Analytic Functions of Several Complex Variables
- Complex Manifolds and Analytic Varieties
- Vector Bundles
- Dualities and Thom Class
- Chern Classes and Localization via Obstruction Theory
- Differential Forms
- Čech–de Rham Cohomology
- Chern–Weil Theory Adapted to Čech–de Rham Cohomology
- Vector Bundles with Metrics and Related Topics
- Localization
- Further Topics
- Residues of Chern Classes on Manifolds
- Residues of Chern Classes on Singular Varieties
- Intersection Product of Complex Subspaces
- Riemann–Roch Theorem
- Appendix A: Commutative Algebra
- Appendix B: Algebraic Topology
Readership: Graduate students and researchers in Algebraic Geometry, Complex Analytic Geometry, Differential Geometry, Topology, Singularity Theory, Complex Dynamical Systems, Algebraic Analysis and Mathematical Physics.
Key Features:
- Simple, clear and precise presentation
- The use of the Chern–Weil theory adapted to the Čech–de Rham cohomology, combined with combinatorial topology, provides a natural way of localizing characteristic classes and producing the associate residues
- The method mentioned in the previous point paves the way for clearer and more precise presentations of the central concepts as well as fundamental and important results that have been treated only globally so far
- The philosophy and techniques will be applicable in other areas of Mathematics to provide novel perspective and results
- Two Appendixes, on Commutative Algebra and Algebraic Topology, are provided to make the book more self-contained
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