Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

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