Several complex variables with connections to algebraic geometry and Lie groups / Joseph L. Taylor.

Format:

Book

Author/Creator:

Taylor, Joseph L., 1941- author.

Series:

Graduate studies in mathematics ; v. 46.

Graduate studies in mathematics, 1065-7339 ; volume 46

Language:

English

Subjects (All):

Functions of several complex variables.

Geometry, Algebraic.

Physical Description:

1 online resource (527 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Chapter 1. Selected Problems in One Complex Variable""; ""Â1.1. Preliminaries""; ""Â1.2. A Simple Problem""; ""Â1.3. Partitions of Unity""; ""Â1.4. The Cauchy-Riemann Equations""; ""Â1.5. The Proof of Proposition 1.2.2""; ""Â1.6. The Mittag-Leffler and Weierstrass Theorems""; ""Â1.7. Conclusions and Comments""; ""Exercises""; ""Chapter 2. Holomorphic Functions of Several Variables""; ""Â2.1. Cauchy's Formula and Power Series Expansions""; ""Â2.2. Hartog's Theorem""; ""Â2.3. The Cauchy-Riemann Equations""

""Â2.4. Convergence Theorems""""Â2.5. Domains of Holomorphy""; ""Exercises""; ""Chapter 3. Local Rings and Varieties""; ""Â3.1. Rings of Germs of Holomorphic Functions""; ""Â3.2. Hilbert's Basis Theorem""; ""Â3.3. The Weierstrass Theorems""; ""Â3.4. The Local Ring of Holomorphic Functions is Noetherian""; ""Â3.5. Varieties""; ""Â3.6. Irreducible Varieties""; ""Â3.7. Implicit and Inverse Mapping Theorems""; ""Â3.8. Holomorphic Functions on a Subvariety""; ""Exercises""; ""Chapter 4. The Nullstellensatz""; ""Â4.1. Reduction to the Case of Prime Ideals""

""Â4.2. Survey of Results on Ring and Field Extensions""""Â4.3. Hilbert's Nullstellensatz""; ""Â4.4. Finite Branched Holomorphic Covers""; ""Â4.5. The Nullstellensatz""; ""Â4.6. Morphisms of Germs of Varieties""; ""Exercises""; ""Chapter 5. Dimension""; ""Â5.1. Topological Dimension""; ""Â5.2. Subvarieties of Codimension 1""; ""Â5.3. Krull Dimension""; ""Â5.4. Tangential Dimension""; ""Â5.5. Dimension and Regularity""; ""Â5.6. Dimension of Algebraic Varieties""; ""Â5.7. Algebraic vs. Holomorphic Dimension""; ""Exercises""; ""Chapter 6. Homological Algebra""

""Â6.1. Abelian Categories""""Â6.2. Complexes""; ""Â6.3. Injective and Projective Resolutions""; ""Â6.4. Higher Derived Functors""; ""Â6.5. Ext""; ""Â6.6. The Category of Modules, Tor""; ""Â6.7. Hilbert's Syzygy Theorem""; ""Exercises""; ""Chapter 7. Sheaves and Sheaf Cohomology""; ""Â7.1. Sheaves""; ""Â7.2. Morphisms of Sheaves""; ""Â7.3. Operations on Sheaves""; ""Â7.4. Sheaf Cohomology""; ""Â7.5. Classes of Acyclic Sheaves""; ""Â7.6. Ringed Spaces""; ""Â7.7. De Rham Cohomology""; ""Â7.8. Ä?ech Cohomology""; ""Â7.9. Line Bundles and ÄŒech Cohomology""; ""Exercises""

""Chapter 8. Coherent Algebraic Sheaves""""Â8.1. Abstract Varieties""; ""Â8.2. Localization""; ""Â8.3. Coherent and Quasi-coherent Algebraic Sheaves""; ""Â8.4. Theorems of Artin-Rees and Krull""; ""Â8.5. The Vanishing Theorem for Quasi-coherent Sheaves""; ""Â8.6. Cohomological Characterization of Affine Varieties""; ""Â8.7. Morphisms - Direct and Inverse Image""; ""Â8.8. An Open Mapping Theorem""; ""Exercises""; ""Chapter 9. Coherent Analytic Sheaves""; ""Â9.1. Coherence in the Analytic Case""; ""Â9.2. Oka's Theorem""; ""Â9.3. Ideal Sheaves""

""Â9.4. Coherent Sheaves on Varieties""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 497-499) and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

9781470420970

147042097X

OCLC:

922964898