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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 32: Several complex variables and analytic spaces

Introduction

Several complex variables is, naturally, the study of (differentiable) functions of more than one complex variable. The rigid constraints imposed by complex differentiability imply that, at least locally, these functions behave almost like polynomials. In particular, study of the related spaces tends to resemble algebraic geometry, except that tools of analysis are used in addition to algebraic constructs. Differential equations on these spaces and automorphisms of them provide useful connections with these other areas.

History

Applications and related fields

For infinite-dimensional holomorphy, See also 46G20, 58B12

Subfields

• 32A: Holomorphic functions of several complex variables
• 32B: Local analytic geometry, see also 13-XX and 14-XX
• 32C: [General theory of] Analytic spaces
• 32D: Analytic continuation
• 32E: Holomorphic convexity
• 32F: Geometric convexity [new in 2000]
• 32G: Deformations of analytic structures
• 32H: Holomorphic mappings and correspondences
• 32J: Compact analytic spaces, For Riemann surfaces, see 14HXX, 30FXX; for algebraic theory, See 14JXX
• 32K: Generalizations of analytic spaces (should also be assigned at least one other classification number in this section)
• 32L: Holomorphic fiber spaces, see also 55RXX
• 32M: Complex spaces with a group of automorphisms
• 32N: Automorphic functions, see also 11FXX, 20H10, 22E40, 30F35, 32P05 Non-Archimedean complex analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)
• 32Q: Complex manifolds [new in 2000]
• 32S: Singularities
• 32T: Pseudoconvex domains [new in 2000]
• 32U: Pluripotential theory [new in 2000]
• 32V: CR Manifolds [new in 2000]
• 32W: Differential operators in several variables [new in 2000]

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Krantz, Steven G.: "What is several complex variables?", Amer. Math. Monthly 94 (1987), no. 3, 236--256. MR88e:32001

Hill, C. Denson: "What is the notion of a complex manifold with a smooth boundary?", Algebraic analysis, Vol. I, 185--201, Academic Press, Boston, MA, 1988. MR90e:32009

"Reviews in Complex Analysis 1980-1986" (four volumes), American Mathematical Society, Providence, RI, 1989. 3064 pp., ISBN 0-8218-0127-9: Reviews reprinted from Mathematical Reviews published during 1980--1986.

Fornæss, John Erik; Stensønes, Berit: "Lectures on counterexamples in several complex variables", Princeton University Press, Princeton, 1987, ISBN 0-691-08456-4

Software and tables

Other web sites with this focus

• Preprint server
• Here are the AMS and Goettingen resource pages for area 32.

Selected topics at this site

• Hartogs' lemma (on removable singularities) in the theory of several complex variables.
• Hartogs' theorem: multivariate analyticity follows from analyticity in each variable
• What's the Edge of the Wedge Theorem?
• Results on analytic functions parallel to those of algebraic geometry
• Zero sets of analytic functions must have appropriate dimensions (Weierstrass Preparation Theorem)
• How to recognize complex varieties among (even-dimensional) real varieties? (That is, can two real equations be treated as one complex equation in half as many complex unknowns?)