**Graduate Studies in Mathematics**

Volume: 46;
2002;
507 pp;
Hardcover

MSC: Primary 34; 14; 22; 43;

Print ISBN: 978-0-8218-3178-6

Product Code: GSM/46

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-2097-0
Product Code: GSM/46.E**

List Price: $89.00

Individual Member Price: $71.20

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# Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

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*Joseph L. Taylor*

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.

#### Table of Contents

# Table of Contents

## Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xiii14 free
- Chapter 1. Selected Problems in One Complex Variable 118 free
- Chapter 2. Holomorphic Functions of Several Variables 2340
- Chapter 3. Local Rings and Varieties 3754
- §3.1. Rings of Germs of Holomorphic Functions 3855
- §3.2. Hilbert's Basis Theorem 3956
- §3.3. The Weierstrass Theorems 4057
- §3.4. The Local Ring of Holomorphic Functions is Noetherian 4461
- §3.5. Varieties 4562
- §3.6. Irreducible Varieties 4966
- §3.7. Implicit and Inverse Mapping Theorems 5067
- §3.8. Holomorphic Functions on a Subvariety 5572
- Exercises 5774

- Chapter 4. The Nullstellensatz 6178
- Chapter 5. Dimension 95112
- Chapter 6. Homological Algebra 113130
- Chapter 7. Sheaves and Sheaf Cohomology 145162
- §7.1. Sheaves 145162
- §7.2. Morphisms of Sheaves 150167
- §7.3. Operations on Sheaves 152169
- §7.4. Sheaf Cohomology 157174
- §7.5. Classes of Acyclic Sheaves 163180
- §7.6. Ringed Spaces 168185
- §7.7. De Rham Cohomology 172189
- §7.8. Ćech Cohomology 174191
- §7.9. Line Bundles and Čech Cohomology 180197
- Exercises 182199

- Chapter 8. Coherent Algebraic Sheaves 185202
- §8.1. Abstract Varieties 186203
- §8.2. Localization 189206
- §8.3. Coherent and Quasi-coherent Algebraic Sheaves 194211
- §8.4. Theorems of Artin-Rees and Krull 197214
- §8.5. The Vanishing Theorem for Quasi-coherent Sheaves 199216
- §8.6. Cohomological Characterization of Affine Varieties 200217
- §8.7. Morphisms - Direct and Inverse Image 204221
- §8.8. An Open Mapping Theorem 207224
- Exercises 212229

- Chapter 9. Coherent Analytic Sheaves 215232
- Chapter 10. Stein Spaces 237254
- Chapter 11. Fréchet Sheaves - Cartan's Theorems 263280
- §11.1. Topological Vector Spaces 264281
- §11.2. The Topology of H(X) 266283
- §11.3. Frechet Sheaves 274291
- §11.4. Cartan's Theorems 277294
- §11.5. Applications of Cartan's Theorems 281298
- §11.6. Invertible Groups and Line Bundles 283300
- §11.7. Meromorphic Functions 284301
- §11.8. Holomorphic Functional Calculus 288305
- §11.9. Localization 298315
- §11.10. Coherent Sheaves on Compact Varieties 300317
- §11.11. Schwartz's Theorem 302319
- Exercises 309326

- Chapter 12. Projective Varieties 313330
- Chapter 13. Algebraic vs. Analytic - Serre's Theorems 331348
- Chapter 14. Lie Groups and Their Representations 357374
- §14.1. Topological Groups 358375
- §14.2. Compact Topological Groups 363380
- §14.3. Lie Groups and Lie Algebras 376393
- §14.4. Lie Algebras 385402
- §14.5. Structure of Semisimple Lie Algebras 392409
- §14.6. Representations of sl[sub(2)](C) 400417
- §14.7. Representations of Semisimple Lie Algebras 404421
- §14.8. Compact Semisimple Groups 409426
- Exercises 416433

- Chapter 15. Algebraic Groups 419436
- §15.1. Algebraic Groups and Their Representations 419436
- §15.2. Quotients and Group Actions 423440
- §15.3. Existence of the Quotient 427444
- §15.4. Jordan Decomposition 430447
- §15.5. Tori 433450
- §15.6. Solvable Algebraic Groups 437454
- §15.7. Semisimple Groups and Borel Subgroups 442459
- §15.8. Complex Semisimple Lie Groups 451468
- Exercises 456473

- Chapter 16. The Borel-Weil-Bott Theorem 459476
- §16.1. Vector Bundles and Induced Representations 460477
- §16.2. Equivariant Line Bundles on the Flag Variety 464481
- §16.3. The Casimir Operator 469486
- §16.4. The Borel-Weil Theorem 474491
- §16.5. The Borel-Weil-Bott Theorem 478495
- §16.6. Consequences for Real Semisimple Lie Groups 483500
- §16.7. Infinite Dimensional Representations 484501
- Exercises 493510

- Bibliography 497514
- Index 501518
- Back Cover Back Cover1527

#### Readership

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

#### 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