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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
Lectures on Counterexamples in Several Complex Variables
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-0-8218-4422-9
Product Code:  CHEL/363.H
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-3120-4
Product Code:  CHEL/363.H.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Hardcover ISBN:  978-0-8218-4422-9
eBook: ISBN:  978-1-4704-3120-4
Product Code:  CHEL/363.H.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
Lectures on Counterexamples in Several Complex Variables
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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
Hardcover ISBN:  978-0-8218-4422-9
Product Code:  CHEL/363.H
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-3120-4
Product Code:  CHEL/363.H.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Hardcover ISBN:  978-0-8218-4422-9
eBook ISBN:  978-1-4704-3120-4
Product Code:  CHEL/363.H.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 363; 1987; 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.
    Readership

    Graduate 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
    • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 363; 1987; 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.
Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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