Contents. Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39.

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1 Preface xiii Chapter 1. Selected Problems in One Complex Variable Preliminaries A Simple Problem Partitions of Unity The Cauchy-Riemann Equations The Proof of Proposition The Mittag-Leffler and Weierstrass Theorems Conclusions and Comments 16 Exercises 18 Chapter 2. Holomorphic Functions of Several Variables Cauchy s Formula and Power Series Expansions Hartog s Theorem The Cauchy-Riemann Equations Convergence Theorems Domains of Holomorphy 31 Exercises 35 Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39 vii

2 viii Contents 3.3. The Weierstrass Theorems The Local Ring of Holomorphic Functions is Noetherian Varieties Irreducible Varieties Implicit and Inverse Mapping Theorems Holomorphic Functions on a Subvariety 55 Exercises 57 Chapter 4. The Nullstellensatz Reduction to the Case of Prime Ideals Survey of Results on Ring and Field Extensions Hilbert s Nullstellensatz Finite Branched Holomorphic Covers The Nullstellensatz Morphisms of Germs of Varieties 87 Exercises 92 Chapter 5. Dimension Topological Dimension Subvarieties of Codimension Krull Dimension Tangential Dimension Dimension and Regularity Dimension of Algebraic Varieties Algebraic vs. Holomorphic Dimension 108 Exercises 110 Chapter 6. Homological Algebra Abelian Categories Complexes Injective and Projective Resolutions Higher Derived Functors Ext The Category of Modules, Tor Hilbert s Syzygy Theorem 137 Exercises 142

3 ix Chapter 7. Sheaves and Sheaf Cohomology Sheaves Morphisms of Sheaves Operations on Sheaves Sheaf Cohomology Classes of Acyclic Sheaves Ringed Spaces De Rham Cohomology Čech Cohomology Line Bundles and Čech Cohomology 180 Exercises 182 Chapter 8. Coherent Algebraic Sheaves Abstract Varieties Localization Coherent and Quasi-coherent Algebraic Sheaves Theorems of Artin-Rees and Krull The Vanishing Theorem for Quasi-coherent Sheaves Cohomological Characterization of Affine Varieties Morphisms Direct and Inverse Image An Open Mapping Theorem 207 Exercises 212 Chapter 9. Coherent Analytic Sheaves Coherence in the Analytic Case Oka s Theorem Ideal Sheaves Coherent Sheaves on Varieties Morphisms between Coherent Sheaves Direct and Inverse Image 229 Exercises 234 Chapter 10. Stein Spaces Dolbeault Cohomology Chains of Syzygies Functional Analysis Preliminaries 245

4 x Contents Cartan s Factorization Lemma Amalgamation of Syzygies Stein Spaces 257 Exercises 260 Chapter 11. Fréchet Sheaves Cartan s Theorems Topological Vector Spaces The Topology of H(X) Fréchet Sheaves Cartan s Theorems Applications of Cartan s Theorems Invertible Groups and Line Bundles Meromorphic Functions Holomorphic Functional Calculus Localization Coherent Sheaves on Compact Varieties Schwartz s Theorem 302 Exercises 309 Chapter 12. Projective Varieties Complex Projective Space Projective Space as an Algebraic and a Holomorphic Variety The Sheaves O(k) and H(k) Applications of the Sheaves O(k) Embeddings in Projective Space 325 Exercises 328 Chapter 13. Algebraic vs. Analytic Serre s Theorems Faithfully Flat Ring Extensions Completion of Local Rings Local Rings of Algebraic vs. Holomorphic Functions The Algebraic to Holomorphic Functor Serre s Theorems Applications 351 Exercises 355

5 xi Chapter 14. Lie Groups and Their Representations Topological Groups Compact Topological Groups Lie Groups and Lie Algebras Lie Algebras Structure of Semisimple Lie Algebras Representations of sl 2 (C) Representations of Semisimple Lie Algebras Compact Semisimple Groups 409 Exercises 416 Chapter 15. Algebraic Groups Algebraic Groups and Their Representations Quotients and Group Actions Existence of the Quotient Jordan Decomposition Tori Solvable Algebraic Groups Semisimple Groups and Borel Subgroups Complex Semisimple Lie Groups 451 Exercises 456 Chapter 16. The Borel-Weil-Bott Theorem Vector Bundles and Induced Representations Equivariant Line Bundles on the Flag Variety The Casimir Operator The Borel-Weil Theorem The Borel-Weil-Bott Theorem Consequences for Real Semisimple Lie Groups Infinite Dimensional Representations 484 Exercises 493 Bibliography 497 Index 501