Several Complex Variables with Connections to Algebraic Geometry and Lie Groups
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...
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PrefaceCh. 1Selected Problems in One Complex Variable11.1Preliminaries21.2A Simple Problem21.3Partitions of Unity41.4The Cauchy-Riemann Equations71.5The Proof of Proposition 1.2.2101.6The Mittag-Leffler and Weierstrass Theorems121.7Conclusions and Comments16Ch. 2Holomorphic Functions of Several Variables232.1Cauchy's Formula and Power Series Expansions232.2Hartog's Theorem262.3The Cauchy-Riemann Equations292.4Convergence Theorems292.5Domains of Holomorphy31Ch. 3Local Rings and Varieties373.1Rings of Germs of Holomorphic Functions383.2Hilbert's Basis Theorem393.3The Weierstrass Theorems403.4The Local Ring of Holomorphic Functions is Noetherian443.5Varieties453.6Irreducible Varieties493.7Implicit and Inverse Mapping Theorems503.8Holomorphic Functions on a Subvariety55Ch. 4The Nullstellensatz614.1Reduction to the Case of Prime Ideals614.2Survey of Results on Ring and Field Extensions624.3Hilbert's Nullstellensatz684.4Finite Branched Holomorphic Covers724.5The Nullstellensatz794.6Morphisms of Germs of Varieties87Ch. 5Dimension955.1Topological Dimension955.2Subvarieties of Codimension 1975.3Krull Dimension995.4Tangential Dimension1005.5Dimension and Regularity1035.6Dimension of Algebraic Varieties1045.7Algebraic vs. Holomorphic Dimension108Ch. 6Homological Algebra1136.1Abelian Categories1136.2Complexes1196.3Injective and Projective Resolutions1226.4Higher Derived Functors1266.5Ext1316.6The Category of Modules, Tor1336.7Hilbert's Syzygy Theorem137Ch. 7Sheaves and Sheaf Cohomology1457.1Sheaves1457.2Morphisms of Sheaves1507.3Operations on Sheaves1527.4Sheaf Cohomology1577.5Classes of Acyclic Sheaves1637.6Ringed Spaces1687.7De Rham Cohomology1727.8Cech Cohomology1747.9Line Bundles and Cech Cohomology180Ch. 8Coherent Algebraic Sheaves1858.1Abstract Varieties1868.2Localization1898.3Coherent and Quasi-coherent Algebraic Sheaves1948.4Theorems of Artin-Rees and Krull1978.5The Vanishing Theorem for Quasi-coherent Sheaves1998.6Cohomological Characterization of Affine Varieties2008.7Morphisms - Direct and Inverse Image2048.8An Open Mapping Theorem207Ch. 9Coherent Analytic Sheaves2159.1Coherence in the Analytic Case2159.2Oka's Theorem2179.3Ideal Sheaves2219.4Coherent Sheaves on Varieties2259.5Morphisms between Coherent Sheaves2269.6Direct and Inverse Image229Ch. 10Stein Spaces23710.1Dolbeault Cohomology23710.2Chains of Syzygies24310.3Functional Analysis Preliminaries24510.4Cartan's Factorization Lemma24810.5Amalgamation of Syzygies25210.6Stein Spaces257Ch. 11Frechet Sheaves - Cartan's Theorems26311.1Topological Vector Spaces26411.2The Topology of H(X)26611.3Frechet Sheaves27411.4Cartan's Theorems27711.5Applications of Cartan's Theorems28111.6Invertible Groups and Line Bundles28311.7Meromorphic Functions28411.8Holomorphic Functional Calculus28811.9Localization29811.10Coherent Sheaves on Compact Varieties30011.11Schwartz's Theorem302Ch. 12Projective Varieties31312.1Complex Projective Space31312.2Projective Space as an Algebraic and a Holomorphic Variety31412.3The Sheaves O(k) and H(k)31712.4Applications of the Sheaves O(k)32312.5Embeddings in Projective Space325Ch. 13Algebraic vs. Analytic - Serre's Theorems33113.1Faithfully Flat Ring Extensions33113.2Completion of Local Rings33413.3Local Rings of Algebraic vs. Holomorphic Functions33813.4The Algebraic to Holomorphic Functor34113.5Serre's Theorems34413.6Applications351Ch. 14Lie Groups and Their Representations35714.1Topological Groups35814.2Compact Topological Groups36314.3Lie Groups and Lie Algebras37614.4Lie Algebras38514.5Structure of Semisimple Lie Algebras39214.6Representations of [actual symbol not reproducible][subscript 2]([Complex number system])40014.7Representations of Semisimple Lie Algebras40414.8Compact Semisimple Groups409Ch. 15Algebraic Groups41915.1Algebraic Groups and Their Representations41915.2Quotients and Group Actions42315.3Existence of the Quotient42715.4Jordan Decomposition43015.5Tori43315.6Solvable Algebraic Groups43715.7Semisimple Groups and Borel Subgroups44215.8Complex Semisimple Lie Groups451Ch. 16The Borel-Weil-Bott Theorem45916.1Vector Bundles and Induced Representations46016.2Equivariant Line Bundles on the Flag Variety46416.3The Casimir Operator46916.4The Borel-Weil Theorem47416.5The Borel-Weil-Bott Theorem47816.6Consequences for Real Semisimple Lie Groups48316.7Infinite Dimensional Representations484Bibliography497Index501
\ From The CriticsMost of the students in the graduate course from which Taylor (U. of Utah) developed this text were planning to specialize either in algebraic geometry or in the representational theory of semisimple Lie groups. He therefore simultaneously develops the fundamentals of both complex algebraic geometry and several complex variables, emphasizing the similarities in techniques of the two at the basic level. Annotation c. Book News, Inc., Portland, OR (booknews.com)\ \
