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Hardcover ISBN:  9780821831786 
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Hardcover ISBN:  9780821831786 
Product Code:  GSM/46 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470420970 
Product Code:  GSM/46.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821831786 
eBook ISBN:  9781470420970 
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 DetailsGraduate Studies in MathematicsVolume: 46; 2002; 507 ppMSC: 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 PeterWeyl 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 BorelWeilBott 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.
ReadershipGraduate 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 BorelWeilBott 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 selfcontained as possible by giving results and proofs of many results from this background material ... This very wellwritten book is a pleasant text for any graduate student and a base for any lecture course on several complex variables.
Mathematical Reviews


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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 PeterWeyl 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 BorelWeilBott 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.
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 BorelWeilBott 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 selfcontained as possible by giving results and proofs of many results from this background material ... This very wellwritten book is a pleasant text for any graduate student and a base for any lecture course on several complex variables.
Mathematical Reviews