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Several Complex Variables with Connections to Algebraic Geometry and Lie Groups
 
Joseph L. Taylor University of Utah, Salt Lake City, UT
Several Complex Variables with Connections to Algebraic Geometry and Lie Groups
Hardcover ISBN:  978-0-8218-3178-6
Product Code:  GSM/46
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-2097-0
Product Code:  GSM/46.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-3178-6
eBook: ISBN:  978-1-4704-2097-0
Product Code:  GSM/46.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
Several Complex Variables with Connections to Algebraic Geometry and Lie Groups
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Several Complex Variables with Connections to Algebraic Geometry and Lie Groups
Joseph L. Taylor University of Utah, Salt Lake City, UT
Hardcover ISBN:  978-0-8218-3178-6
Product Code:  GSM/46
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-2097-0
Product Code:  GSM/46.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-3178-6
eBook ISBN:  978-1-4704-2097-0
Product Code:  GSM/46.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 462002; 507 pp
    MSC: Primary 34; 14; 22; 43;

    This text presents an integrated development of the theory of several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail.

    Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure and representations of complex semisimple Lie groups. Included are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Miličić's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text.

    There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert.

    Readership

    Graduate students and research mathematicians interested in ODEs, algebraic geometry, group theory, generalizations, and abstract harmonic analysis.

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. Selected problems in one complex variable
    • Chapter 2. Holomorphic functions of several variables
    • Chapter 3. Local rings and varieties
    • Chapter 4. The Nullstellensatz
    • Chapter 5. Dimension
    • Chapter 6. Homological algebra
    • Chapter 7. Sheaves and sheaf cohomology
    • Chapter 8. Coherent algebraic sheaves
    • Chapter 9. Coherent analytic sheaves
    • Chapter 10. Stein spaces
    • Chapter 11. Fréchet sheaves—Cartan’s theorems
    • Chapter 12. Projective varieties
    • Chapter 13. Algebraic vs. analytic—Serre’s theorems
    • Chapter 14. Lie groups and their representations
    • Chapter 15. Algebraic groups
    • Chapter 16. The Borel-Weil-Bott theorem
  • Additional Material
     
     
  • Reviews
     
     
    • The book can serve as an excellent text for a graduate course on modern methods of complex analysis, as well as a useful reference for those working in analysis.

      Zentralblatt MATH
    • The author succeeded in making the text as self-contained as possible by giving results and proofs of many results from this background material ... This very well-written book is a pleasant text for any graduate student and a base for any lecture course on several complex variables.

      Mathematical Reviews
  • 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
Volume: 462002; 507 pp
MSC: Primary 34; 14; 22; 43;

This text presents an integrated development of the theory of several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail.

Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure and representations of complex semisimple Lie groups. Included are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Miličić's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text.

There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert.

Readership

Graduate students and research mathematicians interested in ODEs, algebraic geometry, group theory, generalizations, and abstract harmonic analysis.

  • Chapters
  • Chapter 1. Selected problems in one complex variable
  • Chapter 2. Holomorphic functions of several variables
  • Chapter 3. Local rings and varieties
  • Chapter 4. The Nullstellensatz
  • Chapter 5. Dimension
  • Chapter 6. Homological algebra
  • Chapter 7. Sheaves and sheaf cohomology
  • Chapter 8. Coherent algebraic sheaves
  • Chapter 9. Coherent analytic sheaves
  • Chapter 10. Stein spaces
  • Chapter 11. Fréchet sheaves—Cartan’s theorems
  • Chapter 12. Projective varieties
  • Chapter 13. Algebraic vs. analytic—Serre’s theorems
  • Chapter 14. Lie groups and their representations
  • Chapter 15. Algebraic groups
  • Chapter 16. The Borel-Weil-Bott theorem
  • The book can serve as an excellent text for a graduate course on modern methods of complex analysis, as well as a useful reference for those working in analysis.

    Zentralblatt MATH
  • The author succeeded in making the text as self-contained as possible by giving results and proofs of many results from this background material ... This very well-written book is a pleasant text for any graduate student and a base for any lecture course on several complex variables.

    Mathematical Reviews
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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