Several Complex Variables
A branch of mathematics studying holomorphic functions of multiple complex variables, extending complex analysis to higher dimensions with distinct geometric and analytic properties.
Several Complex Variables
Several complex variables (SCV) is a fundamental area of mathematics that extends complex analysis from functions of a single complex variable to functions of multiple complex variables. This field emerges at the intersection of complex analysis, algebraic geometry, and differential geometry, bringing together powerful techniques from each domain.
Fundamental Concepts
Domain Structure
Unlike in single-variable complex analysis, the domains in SCV have richer structure:
- Polydiscs - direct products of discs in each variable
- Domain of holomorphy - domains where holomorphic functions achieve their maximum on the boundary
- Pseudoconvex domains - natural domains for holomorphic functions
Key Properties
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Hartogs' Phenomenon: Unlike single-variable complex analysis, a function holomorphic on a punctured domain in several variables can often be extended across the puncture, demonstrating the rigidity of several complex variables.
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Cauchy-Riemann equations generalize to multiple variables, yielding systems of partial differential equations that characterize holomorphic functions.
Major Theorems
Core Results
- Cartan's Theorems A and B - describing coherent sheaves
- Oka's Coherence Theorem - fundamental for analytic sheaves
- Hartogs' Extension Theorem - on removable singularities
Applications
The theory finds applications in:
Historical Development
The field emerged in the early 20th century through work of:
- Friedrich Hartogs
- Henri Cartan
- Kiyoshi Oka
- Karl Weierstrass
Modern Developments
Current research directions include:
- ∂-equations and their applications
- Complex dynamics in several variables
- Connection to mirror symmetry
Relationship to Other Areas
SCV maintains deep connections with:
The field continues to evolve, providing essential tools for modern mathematics and theoretical physics, while generating new questions and approaches to classical problems.