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.
Author(s): Joseph L. Taylor
Series: Graduate Studies in Mathematics, V. 46
Publisher: American Mathematical Society
Year: 2002
Language: English
Pages: xvi+507
Preface
Chapter 1 Selected Problems in One Complex Variable
1.1 Preliminaries
1.2 A Simple Problem
1.3 Partitions of Unity
1.4 The Cauchy-Riemann Equations
1.5 The Proof of Proposition 1.2.2.
1.6 The Mittag-Leffler and Weierstrass Theorems
1.7 Conclusions and Comments
Exercises
Chapter 2 Holomorphic Functions of Several Variables
2.1 Cauchy's Formula and Power Series Expansions
2.2 Hartog's Theorem
2.3 The Cauchy-Riemann Equations
2.4 Convergence Theorems
2.5 Domains of Holomorphy
Exercises
Chapter 3 Local Rings and Varieties
3.1 Rings of Germs of Holomorphic Functions
3.2 Hilbert's Basis Theorem
3.3 The Weierstrass Theorems
3.4 The Local Ring of Holomorphic Functions Is Noetherian
3.5 Varieties
3.6 Irreducible Varieties
3.7 Implicit and Inverse Mapping Theorems
3.8 Holomorphic ]Functions on a Subvariety
Exercises
Chapter 4 The Nullstellensatz
4.1 Reduction to the Case of Prime Ideals
4.2 Survey of Results on Ring and Field Extensions
4.3 Hilbert's Nullstellensatz
4.4 Finite Branched Holomorphic Covers
4.5 The Nullstellensatz
4.6 Morphisms of Germs of Varieties
Exercises
Chapter 5 Dimension
5.1 Topological Dimension
5.2 Subvarieties of Codimension 1
5.3 Krull Dimension
5.4 Tangential Dimension
5.5 Dimension and Regularity
5.6 Dimension of Algebraic Varieties
5.7 Algebraic vs. Holomorphic Dimension
Exercises
Chapter 6 Homological Algebra
6.1 Abelian Categories
6.2 Complexes
6.3 Injective and Projective Resolutions
6.4 Higher Derived Functors
6.5 Ext
6.6 The Category of Modules, Tor
6.7 Hilbert's Syzygy Theorem
Exercises
Chapter 7 Sheaves and Sheaf Cohomology
7.1 Sheaves
7.2 Morphisms of Sheaves
7.3 Operations on Sheaves
7.4 Sheaf Cohomology
7.5 Classes of Acyclic Sheaves
7.6 Ringed Spaces
7.7 De Rham Cohomology
7.8 Cech Cohomology
7.9 Line Bundles and Cech Cohomology
Exercises
Chapter 8 Coherent Algebraic Sheaves
8.1 Abstract Varieties
8.2 Localization
8.3 Coherent and Quasi-coherent Algebraic Sheaves
8.4 Theorems of Artin-Rees and Krull
8.5 The Vanishing Theorem for Quasi-coherent Sheaves
8.6 Cohomological Characterization of Affine Varieties
8.7 Morphisms - Direct and Inverse Image
8.8 An Open Mapping Theorem
Exercises
Chapter 9 Coherent Analytic Sheaves
9.1 Coherence in the Analytic Case
9.2 Oka's Theorem
9.3 Ideal Sheaves
9.4 Coherent Sheaves on Varieties
9.5 Morphisms between Coherent Sheaves
9.6 Direct and Inverse Image
Exercises
Chapter 10 Stein Spaces
10.1 Dolbeault Cohomology
10.2 Chains of Syzygies
10.3 Functional Analysis Preliminaries
10.4 Cartan's Factorization Lemma
10.5 Amalgamation of Syzygies
10.6 Stein Spaces
Exercises
Chapter 11 Frechet Sheaves -Cartan's Theorems
11.1 Topological Vector Spaces
11.2 The Topology of H(X)
11.3 Frechet Sheaves
11.4 Cartan's Theorems
11.5 Applications of Cartan's Theorems
11.6 Invertible Groups and Line Bundles
11.7 Meromorphic ]Functions
11.8 Holomorphic Functional Calculus
11.9 Localization
11.10 Coherent Sheaves on Compact Varieties
11.11 Schwartz's Theorem
Exercises
Chapter 12 Projective Varieties
12.1 Complex Projective Space
12.2 Projective Space as an Algebraic and a Holomorphic Variety
12.3 The Sheaves 0(k) and H(k)
12.4 Applications of the Sheaves 0(k)
12.5 Embeddings in Projective Space
Exercises
Chapter 13 Algebraic vs. Analytic -Serre's Theorems
13.1 Faithfully Flat Ring Extensions
13.2 Completion of Local Rings
13.3 Local Rings of Algebraic vs. Holomorphic Functions
13.4 The Algebraic to Holomorphic Functor
13.5 Serre's Theorems
13.6 Applications
Exercises
Chapter 14 Lie Groups and Their Representations
14.1 Topological Groups
14.2 Compact Topological Groups
14.3 Lie Groups and Lie Algebras
14.4 Lie Algebras
14.5 Structure of Semisimple Lie Algebras
14.6 Representations of sl2(C)
14.7 Representations of Semisimple Lie Algebras
14.8 Compact Semisimple Groups
Exercises
Chapter 15 Algebraic Groups
15.1 Algebraic Groups and Their Representations
15.2 Quotients and Group Actions
15.3 Existence of the Quotient
15.4 Jordan Decomposition
15.5 Tori
15.6 Solvable Algebraic Groups
15.7 Semisimple Groups and Borel Subgroups
15.8 Complex Semisimple Lie Groups
Exercises
Chapter 16 The Borel-Weil-Bott Theorem
16.1 Vector Bundles and Induced Representations
16.2 Equivariant Line Bundles on the Flag Variety
16.3 The Casimir Operator
16.4 The Borel-Weil Theorem
16.5 The Borel-Weil-Bott Theorem
16.6 Consequences for Real Semisimple Lie Groups
16.7 Infinite Dimensional Representations
Exercises
Bibliography
Index
