This volume studies functions of several complex variables, complex manifolds, and analytic spaces.
This volume studies functions of several complex variables, complex manifolds, and analytic spaces. It extends complex analysis into higher dimensions, where geometry and sheaf methods become central.
Part I. Functions of Several Complex Variables
Chapter 1. Complex Euclidean Space
1.1 Coordinates in ℂⁿ 1.2 Domains and open sets 1.3 Holomorphic functions 1.4 Complex differentiability 1.5 Examples
Chapter 2. Power Series
2.1 Several-variable power series 2.2 Domains of convergence 2.3 Holomorphic germs 2.4 Analytic continuation 2.5 Examples
Chapter 3. Cauchy Theory
3.1 Cauchy integral formula in several variables 3.2 Polydiscs 3.3 Hartogs phenomenon 3.4 Bochner–Martinelli formula 3.5 Applications
Part II. Domains of Holomorphy
Chapter 4. Pseudoconvexity
4.1 Convexity vs pseudoconvexity 4.2 Plurisubharmonic functions 4.3 Levi form 4.4 Examples 4.5 Applications
Chapter 5. Domains of Holomorphy
5.1 Definition 5.2 Holomorphic convexity 5.3 Envelopes of holomorphy 5.4 Cartan–Thullen theorem 5.5 Examples
Chapter 6. Stein Manifolds
6.1 Definition and examples 6.2 Cartan theorems A and B 6.3 Coherent analytic sheaves 6.4 Embedding theorems 6.5 Applications
Part III. Analytic Sets and Spaces
Chapter 7. Analytic Sets
7.1 Local zero sets 7.2 Irreducible components 7.3 Regular and singular points 7.4 Dimension 7.5 Examples
Chapter 8. Analytic Spaces
8.1 Structure sheaves 8.2 Local rings 8.3 Morphisms 8.4 Reduced spaces 8.5 Examples
Chapter 9. Singularities
9.1 Isolated singularities 9.2 Normalization 9.3 Resolution overview 9.4 Local invariants 9.5 Applications
Part IV. Sheaves and Cohomology
Chapter 10. Sheaves in Complex Analysis
10.1 Presheaves and sheaves 10.2 Sheaf of holomorphic functions 10.3 Coherent sheaves 10.4 Exact sequences 10.5 Examples
Chapter 11. Cohomology
11.1 Čech cohomology 11.2 Dolbeault cohomology 11.3 De Rham connections 11.4 Vanishing theorems 11.5 Applications
Chapter 12. The ∂̄ Problem
12.1 Differential forms 12.2 Dolbeault operator 12.3 Solving ∂̄ equations 12.4 L² methods overview 12.5 Applications
Part V. Complex Manifolds
Chapter 13. Complex Manifolds
13.1 Definitions 13.2 Holomorphic maps 13.3 Tangent and cotangent bundles 13.4 Examples 13.5 Basic properties
Chapter 14. Hermitian and Kähler Geometry
14.1 Hermitian metrics 14.2 Kähler forms 14.3 Connections and curvature 14.4 Examples 14.5 Applications
Chapter 15. Compact Complex Manifolds
15.1 Compactness phenomena 15.2 Meromorphic functions 15.3 Divisors and line bundles 15.4 Hodge theory overview 15.5 Examples
Part VI. Connections to Algebraic Geometry
Chapter 16. Complex Algebraic Varieties
16.1 Algebraic vs analytic sets 16.2 GAGA overview 16.3 Projective varieties 16.4 Analytic interpretation 16.5 Examples
Chapter 17. Moduli and Deformation
17.1 Deformations of complex structures 17.2 Kodaira–Spencer theory overview 17.3 Moduli spaces 17.4 Obstruction theory 17.5 Applications
Chapter 18. Complex Spaces and Schemes
18.1 Analytification 18.2 Proper maps 18.3 Coherent sheaves 18.4 Comparison results 18.5 Applications
Part VII. Applications and Computation
Chapter 19. Complex Dynamics
19.1 Holomorphic maps in several variables 19.2 Iteration and stability 19.3 Julia-type sets overview 19.4 Dynamical degrees 19.5 Applications
Chapter 20. PDE and Analysis Applications
20.1 Elliptic regularity 20.2 Boundary problems 20.3 Bergman kernels 20.4 Several-variable integral operators 20.5 Applications
Chapter 21. Computational Aspects
21.1 Local analytic computation 21.2 Power series algorithms 21.3 Symbolic methods 21.4 Numerical complex geometry 21.5 Software tools
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Hodge theory 22.2 Birational complex geometry 22.3 Non-Kähler geometry 22.4 Complex Monge–Ampère equations 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification of complex manifolds 23.2 Singularities and resolutions 23.3 Boundary regularity 23.4 Moduli questions 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of several complex variables 24.2 Key contributors 24.3 Evolution of analytic spaces 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Several-variable notation reference B. Key theorems summary C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume develops the higher-dimensional theory of complex analysis. It emphasizes the shift from one-variable function theory to geometry, sheaves, and analytic spaces.