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Cover Art
PRINTED BOOKS
Author Taylor, Joseph L., 1941-

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

Published Providence, R.I. : American Mathematical Society, [2002]
©2002

Copies

Location Call No. Status
 UniM ERC  515.94 TAYL    AVAILABLE
Physical description xvi, 507 pages ; 26 cm.
Series Graduate studies in mathematics, 1065-7339 ; 46.
Graduate studies in mathematics. 1065-7339 ; 46.
Bibliography Includes bibliographical references and index.
Contents Chapter 1 Selected Problems in One Complex Variable 1 -- 1.3 Partitions of Unity 4 -- 1.4 Cauchy-Riemann Equations 7 -- 1.5 Proof of Proposition 1.2.2 10 -- 1.6 Mittag-Leffler and Weierstrass Theorems 12 -- Chapter 2 Holomorphic Functions of Several Variables 23 -- 2.1 Cauchy's Formula and Power Series Expansions 23 -- 2.2 Hartog's Theorem 26 -- 2.3 Cauchy-Riemann Equations 29 -- 2.4 Convergence Theorems 29 -- 2.5 Domains of Holomorphy 31 -- Chapter 3 Local Rings and Varieties 37 -- 3.1 Rings of Germs of Holomorphic Functions 38 -- 3.2 Hilbert's Basis Theorem 39 -- 3.3 Weierstrass Theorems 40 -- 3.4 Local Ring of Holomorphic Functions is Noetherian 44 -- 3.5 Varieties 45 -- 3.6 Irreducible Varieties 49 -- 3.7 Implicit and Inverse Mapping Theorems 50 -- 3.8 Holomorphic Functions on a Subvariety 55 -- Chapter 4 Nullstellensatz 61 -- 4.1 Reduction to the Case of Prime Ideals 61 -- 4.2 Survey of Results on Ring and Field Extensions 62 -- 4.3 Hilbert's Nullstellensatz 68 -- 4.4 Finite Branched Holomorphic Covers 72 -- 4.5 Nullstellensatz 79 -- 4.6 Morphisms of Germs of Varieties 87 -- Chapter 5 Dimension 95 -- 5.1 Topological Dimension 95 -- 5.2 Subvarieties of Codimension 1 97 -- 5.3 Krull Dimension 99 -- 5.4 Tangential Dimension 100 -- 5.5 Dimension and Regularity 103 -- 5.6 Dimension of Algebraic Varieties 104 -- 5.7 Algebraic vs. Holomorphic Dimension 108 -- Chapter 6 Homological Algebra 113 -- 6.1 Abelian Categories 113 -- 6.2 Complexes 119 -- 6.3 Injective and Projective Resolutions 122 -- 6.4 Higher Derived Functors 126 -- 6.5 Ext 131 -- 6.6 Category of Modules, Tor 133 -- 6.7 Hilbert's Syzygy Theorem 137 -- Chapter 7 Sheaves and Sheaf Cohomology 145 -- 7.1 Sheaves 145 -- 7.2 Morphisms of Sheaves 150 -- 7.3 Operations on Sheaves 152 -- 7.4 Sheaf Cohomology 157 -- 7.5 Classes of Acyclic Sheaves 163 -- 7.6 Ringed Spaces 168 -- 7.7 De Rham Cohomology 172 -- 7.8 Cech Cohomology 174 -- 7.9 Line Bundles and Cech Cohomology 180 -- Chapter 8 Coherent Algebraic Sheaves 185 -- 8.1 Abstract Varieties 186 -- 8.2 Localization 189 -- 8.3 Coherent and Quasi-coherent Algebraic Sheaves 194 -- 8.4 Theorems of Artin-Rees and Krull 197 -- 8.5 Vanishing Theorem for Quasi-coherent Sheaves 199 -- 8.6 Cohomological Characterization of Affine Varieties 200 -- 8.7 Morphisms-Direct and Inverse Image 204 -- 8.8 An Open Mapping Theorem 207 -- Chapter 9 Coherent Analytic Sheaves 215 -- 9.1 Coherence in the Analytic Case 215 -- 9.2 Oka's Theorem 217 -- 9.3 Ideal Sheaves 221 -- 9.4 Coherent Sheaves on Varieties 225 -- 9.5 Morphisms between Coherent Sheaves 226 -- 9.6 Direct and Inverse Image 229 -- Chapter 10 Stein Spaces 237 -- 10.1 Dolbeault Cohomology 237 -- 10.2 Chains of Syzygies 243 -- 10.3 Functional Analysis Preliminaries 245 -- 10.4 Cartan's Factorization Lemma 248 -- 10.5 Amalgamation of Syzygies 252 -- 10.6 Stein Spaces 257 -- Chapter 11 Frechet Sheaves-Cartan's Theorems 263 -- 11.1 Topological Vector Spaces 264 -- 11.2 Topology of H(X) 266 -- 11.3 Frechet Sheaves 274 -- 11.4 Cartan's Theorems 277 -- 11.5 Applications of Cartan's Theorems 281 -- 11.6 Invertible Groups and Line Bundles 283 -- 11.7 Meromorphic Functions 284 -- 11.8 Holomorphic Functional Calculus 288 -- 11.9 Localization 298 -- 11.10 Coherent Sheaves on Compact Varieties 300 -- 11.11 Schwartz's Theorem 302 -- Chapter 12 Projective Varieties 313 -- 12.1 Complex Projective Space 313 -- 12.2 Projective Space as an Algebraic and a Holomorphic Variety 314 -- 12.3 Sheaves O(k) and H(k) 317 -- 12.4 Applications of the Sheaves O(k) 323 -- 12.5 Embeddings in Projective Space 325 -- Chapter 13 Algebraic vs. Analytic-Serre's Theorems 331 -- 13.1 Faithfully Flat Ring Extensions 331 -- 13.2 Completion of Local Rings 334 -- 13.3 Local Rings of Algebraic vs. Holomorphic Functions 338 -- 13.4 Algebraic to Holomorphic Functor 341 -- 13.5 Serre's Theorems 344 -- 13.6 Applications 351 -- Chapter 14 Lie Groups and Their Representations 357 -- 14.1 Topological Groups 358 -- 14.2 Compact Topological Groups 363 -- 14.3 Lie Groups and Lie Algebras 376 -- 14.4 Lie Algebras 385 -- 14.5 Structure of Semisimple Lie Algebras 392 -- 14.6 Representations of sl[subscript]2 (C) 400 -- 14.7 Representations of Semisimple Lie Algebras 404 -- 14.8 Compact Semisimple Groups 409 -- Chapter 15 Algebraic Groups 419 -- 15.1 Algebraic Groups and Their Representations 419 -- 15.2 Quotients and Group Actions 423 -- 15.3 Existence of the Quotient 427 -- 15.4 Jordan Decomposition 430 -- 15.5 Tori 433 -- 15.6 Solvable Algebraic Groups 437 -- 15.7 Semisimple Groups and Borel Subgroups 442 -- 15.8 Complex Semisimple Lie Groups 451 -- Chapter 16 Borel-Weil-Bott Theorem 459 -- 16.1 Vector Bundles and Induced Representations 460 -- 16.2 Equivariant Line Bundles on the Flag Variety 464 -- 16.3 Casimir Operator 469 -- 16.4 Borel-Weil Theorem 474 -- 16.5 Borel-Weil-Bott Theorem 478 -- 16.6 Consequences for Real Semisimple Lie Groups 483 -- 16.7 Infinite Dimensional Representations 484.
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 Milicic's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text."--Publisher's website.
Subject Functions of several complex variables.
Geometry, Algebraic.
Lie groups.
ISBN 9780821831786 (hbk. : acid-free paper)
082183178X (hbk. : acid-free paper) £45.00