This introduction to the theory of complex manifolds covers the most important branches and methods in complex analysis of several variables while completely avoiding abstract concepts involving sheaves, coherence, and higher-dimensional cohomology. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Each chapter contains a variety of examples and exercises.

InhaltsangabeI Holomorphic Functions.- 1. Complex Geometry.- Real and Complex Structures.- Hermitian Forms and Inner Products.- Balls and Polydisks.- Connectedness.- Reinhardt Domains.- 2. Power Series.- Polynomials.- Convergence.- Power Series.- 3. Complex Differentiable Functions.- The Complex Gradient.- Weakly Holomorphic Functions.- Holomorphic Functions.- 4. The Cauchy Integral.- The Integral Formula.- Holomorphy of the Derivatives.- The Identity Theorem.- 5. The Hartogs Figure.- Expansion in Reinhardt Domains.- Hartogs Figures.- 6. The Cauchy-Riemann Equations.- Real Differentiable Functions.- Wirtinger's Calculus.- The Cauchy-Riemann Equations.- 7. Holomorphic Maps.- The Jacobian.- Chain Rules.- Tangent Vectors.- The Inverse Mapping.- 8. Analytic Sets.- Analytic Subsets.- Bounded Holomorphic Functions.- Regular Points.- Injective Holomorphic Mappings.- II Domains of Holomorphy.- 1. The Continuity Theorem.- General Hartogs Figures.- Removable Singularities.- The Continuity Principle.- Hartogs Convexity.- Domains of Holomorphy.- 2. Plurisubharmonic Functions.- Subharmonic Functions.- The Maximum Principle.- Differentiate Subharmonic Functions.- Plurisubharmonic Functions.- The Levi Form.- Exhaustion Functions.- 3. Pseudoconvexity.- Pseudoconvexity.- The Boundary Distance.- Properties of Pseudoconvex Domains.- 4. Levi Convex Boundaries.- Boundary Functions.- The Levi Condition.- Affine Convexity.- A Theorem of Levi.- 5. Holomorphic Convexity.- Affine Convexity.- Holomorphic Convexity.- The Cartan-Thullen Theorem.- 6. Singular Functions.- Normal Exhaustions.- Unbounded Holomorphic Functions.- Sequences.- 7. Examples and Applications.- Domains of Holomorphy.- Complete Reinhardt Domains.- Analytic Polyhedra.- 8. Riemann Domains over Cn.- Riemann Domains.- Union of Riemann Domains.- 9. The Envelope of Holomorphy.- Holomorphy on Riemann Domains.- Envelopes of Holomorphy.- Pseudoconvexity.- Boundary Points.- Analytic Disks.- III Analytic Sets.- 1. The Algebra of Power Series.- The Banach Algebra Bt.- Expansion with Respect to z1.- Convergent Series in Banach Algebras.- Convergent Power Series.- Distinguished Directions.- 2. The Preparation Theorem.- Division with Remainder in Bt.- The Weierstrass Condition.- Weierstrass Polynomials.- Weierstrass Preparation Theorem.- 3. Prime Factorization.- Unique Factorization.- Gauss's Lemma.- Factorization in Hn.- Hensel's Lemma.- The Noetherian Property.- 4. Branched Coverings.- Germs.- Pseudopolynomials.- Euclidean Domains.- The Algebraic Derivative.- Symmetric Polynomials.- The Discriminant.- Hypersurfaces.- The Unbranched Part.- Decompositions.- Projections.- 5. Irreducible Components.- Embedded-Analytic Sets.- Images of Embedded-Analytic Sets.- Local Decomposition.- Analyticity.- The Zariski Topology.- Global Decompositions.- 6. Regular and Singular Points.- Compact Analytic Sets.- Embedding of Analytic Sets.- Regular Points of an Analytic Set.- The Singular Locus.- Extending Analytic Sets.- The Local Dimension.- IV Complex Manifolds.- 1. The Complex Structure.- Complex Coordinates.- Holomorphic Functions.- Riemann Surfaces.- Holomorphic Mappings.- Cartesian Products.- Analytic Subsets.- Differentiable Functions.- Tangent Vectors.- The Complex Structure on the Space of Derivations.- The Induced Mapping.- Immersions and Submersions.- Gluing.- 2. Complex Fiber Bundles.- Lie Groups and Transformation Groups.- Fiber Bundles.- Equivalence.- Complex Vector Bundles.- Standard Constructions.- Lifting of Bundles.- Subbundles and Quotients.- 3. Cohomology.- Cohomology Groups.- Refinements.- Acyclic Coverings.- Generalizations.- The Singular Cohomology.- 4. Meromorphie Functions and Divisors.- The Ring of Germs.- Analytic Hypersurfaces.- Meromorphic Functions.- Divisors.- Associated Line Bundles.- Meromorphic Sections.- 5. Quotients and Submanifolds.- Topological Quotients.- Analytic Decompositions.- Properly Discontinuously Acting Groups.- Complex Tori.- Hopf Manifolds.- The Complex Projective Space.- Meromor

Leseprobe

* Holomorphic Functions * Domains of Holomorphy * Analytic Sets * Complex Manifolds * Stein Theory * Kaehler Manifolds * Boundary Behavior