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Lectures on Counterexamples in Several Complex Variables

John Erik Fornæss University of Michigan, Ann Arbor, Ann Arbor, MI
Berit Stensønes University of Michigan, Ann Arbor, Ann Arbor, MI
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
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Hardcover ISBN: 978-0-8218-4422-9
Product Code: CHEL/363.H
List Price: $47.00 MAA Member Price:$42.30
AMS Member Price: $42.30 Electronic ISBN: 978-1-4704-3120-4 Product Code: CHEL/363.H.E List Price:$44.00
MAA Member Price: $39.60 AMS Member Price:$39.60
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AMS Member Price: $63.45 Click above image for expanded view Lectures on Counterexamples in Several Complex Variables John Erik Fornæss University of Michigan, Ann Arbor, Ann Arbor, MI Berit Stensønes University of Michigan, Ann Arbor, Ann Arbor, MI AMS Chelsea Publishing: An Imprint of the American Mathematical Society Available Formats:  Hardcover ISBN: 978-0-8218-4422-9 Product Code: CHEL/363.H  List Price:$47.00 MAA Member Price: $42.30 AMS Member Price:$42.30
 Electronic ISBN: 978-1-4704-3120-4 Product Code: CHEL/363.H.E
 List Price: $44.00 MAA Member Price:$39.60 AMS Member Price: $39.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$70.50 MAA Member Price: $63.45 AMS Member Price:$63.45
• Book Details

AMS Chelsea Publishing
Volume: 3631987; 247 pp
MSC: 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 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.

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
• CR-manifolds
• Pseudoconvex domains without pseudoconvex exhaustion
• Stein neighborhood basis
• Riemann domains over $\mathbb {C}^n$
• The Kohn-Nirenberg example
• Peak points
• Bloom’s example
• D’Angelo’s example
• Integral manifolds
• Peak sets for A(D)
• Peak sets. Steps 1–4
• Sup-norm 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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Volume: 3631987; 247 pp
MSC: 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 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.

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
• CR-manifolds
• Pseudoconvex domains without pseudoconvex exhaustion
• Stein neighborhood basis
• Riemann domains over $\mathbb {C}^n$
• The Kohn-Nirenberg example
• Peak points
• Bloom’s example
• D’Angelo’s example
• Integral manifolds
• Peak sets for A(D)
• Peak sets. Steps 1–4
• Sup-norm 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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