**AMS Chelsea Publishing**

Volume: 363;
1987;
247 pp;
Hardcover

MSC: Primary 32;

Print ISBN: 978-0-8218-4422-9

Product Code: CHEL/363.H

List Price: $44.00

Individual Member Price: $39.60

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**Electronic ISBN: 978-1-4704-3120-4
Product Code: CHEL/363.H.E**

List Price: $44.00

Individual Member Price: $39.60

# Lectures on Counterexamples in Several Complex Variables

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*John Erik Fornæss; Berit Stensønes*

Counterexamples are remarkably effective for understanding the meaning,
and the limitations, of mathematical results. Fornæss and
Stensønes
look at some of the major ideas of several complex variables by
considering counterexamples to what might seem like reasonable variations
or generalizations. The first part of the book reviews some of the basics
of the theory, in a self-contained introduction to several complex
variables. The counterexamples cover a variety of important topics: the
Levi problem, plurisubharmonic functions, Monge-Ampère equations, CR
geometry, function theory, and the \(\bar\partial\) equation.

The book would be an excellent supplement to a graduate course on several
complex variables.

#### Readership

Graduate students and research mathematicians interested in several complex variables.

#### Table of Contents

# Table of Contents

## Lectures on Counterexamples in Several Complex Variables

- Cover Cover11
- Title page i2
- Contents iii4
- Introduction vii8
- Lectures on counterexamples in several complex variables 110
- Some notations and definitions 211
- Holomorphic functions 615
- Holomorphic convexity and domains of holomorphy 1120
- Stein manifolds 1726
- Subharmonic/Plurisubharmonic functions 2130
- Pseudoconvex domains 4251
- Invariant metrics 5564
- Biholomorphic maps 6069
- Counterexamples to smoothing of plurisubharmonic functions 6675
- Complex Monge Ampère equation 8392
- 𝐻^{∞}-convexity 8796
- CR-manifolds 91100
- Pseudoconvex domains without pseudoconvex exhaustion 102111
- Stein neighborhood basis 105114
- Riemann domains over ℂⁿ 115124
- The Kohn-Nirenberg example 119128
- Peak points 123132
- Bloom’s example 126135
- D’Angelo’s example 129138
- Integral manifolds 133142
- Peak sets for A(D) 138147
- Peak sets. Steps 1–4 141150
- Sup-norm estimates for the ∂-equation 165174
- Sibony’s ∂-example 168177
- Hypoellipticity for ∂ 174183
- Inner functions 178187
- Large maximum modulus sets 189198
- Zero sets 194203
- Nontangential boundary limits of functions in 𝐻^{∞}(𝔹ⁿ) 202211
- Wermer’s example 212221
- The union problem 214223
- Riemann domains 218227
- Runge exhaustion 222231
- Peak sets in weakly pseudoconvex boundaries 229238
- The Kobayashi metric 236245
- Bibliography 242251
- Back Cover Back Cover1260