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Product Code:  CHEL/363.H 
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Electronic ISBN:  9781470431204 
Product Code:  CHEL/363.H.E 
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Book DetailsAMS Chelsea PublishingVolume: 363; 1987; 247 ppMSC: Primary 32;
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 selfcontained introduction to several complex variables. The counterexamples cover a variety of important topics: the Levi problem, plurisubharmonic functions, MongeAmpère 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.ReadershipGraduate students and research mathematicians interested in several complex variables.

Table of Contents

Lectures on counterexamples in several complex variables

Some notations and definitions

Holomorphic functions

Holomorphic convexity and domains of holomorphy

Stein manifolds

Subharmonic/Plurisubharmonic functions

Pseudoconvex domains

Invariant metrics

Biholomorphic maps

Counterexamples to smoothing of plurisubharmonic functions

Complex Monge Ampère equation

$H^\infty $convexity

CRmanifolds

Pseudoconvex domains without pseudoconvex exhaustion

Stein neighborhood basis

Riemann domains over $\mathbb {C}^n$

The KohnNirenberg example

Peak points

Bloom’s example

D’Angelo’s example

Integral manifolds

Peak sets for A(D)

Peak sets. Steps 1–4

Supnorm estimates for the $\bar {\partial }$equation

Sibony’s $\bar {\partial }$example

Hypoellipticity for $\bar {\partial }$

Inner functions

Large maximum modulus sets

Zero sets

Nontangential boundary limits of functions in $H^\infty (\mathbb {B}^n)$

Wermer’s example

The union problem

Riemann domains

Runge exhaustion

Peak sets in weakly pseudoconvex boundaries

The Kobayashi metric


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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 selfcontained introduction to several complex variables. The counterexamples cover a variety of important topics: the Levi problem, plurisubharmonic functions, MongeAmpère 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.
Graduate students and research mathematicians interested in several complex variables.

Lectures on counterexamples in several complex variables

Some notations and definitions

Holomorphic functions

Holomorphic convexity and domains of holomorphy

Stein manifolds

Subharmonic/Plurisubharmonic functions

Pseudoconvex domains

Invariant metrics

Biholomorphic maps

Counterexamples to smoothing of plurisubharmonic functions

Complex Monge Ampère equation

$H^\infty $convexity

CRmanifolds

Pseudoconvex domains without pseudoconvex exhaustion

Stein neighborhood basis

Riemann domains over $\mathbb {C}^n$

The KohnNirenberg example

Peak points

Bloom’s example

D’Angelo’s example

Integral manifolds

Peak sets for A(D)

Peak sets. Steps 1–4

Supnorm estimates for the $\bar {\partial }$equation

Sibony’s $\bar {\partial }$example

Hypoellipticity for $\bar {\partial }$

Inner functions

Large maximum modulus sets

Zero sets

Nontangential boundary limits of functions in $H^\infty (\mathbb {B}^n)$

Wermer’s example

The union problem

Riemann domains

Runge exhaustion

Peak sets in weakly pseudoconvex boundaries

The Kobayashi metric