This book is an introduction to the theory of complex manifolds. The author's intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. Each chapter is complemented by a variety of examples and exercises. The only prerequisite needed to read this book is a knowledge of real analysis and some basic facts from algebra, topology, and the theory of one complex variable. The book can be used as a first introduction to several complex variables as well as a reference for the expert.
Klaus Fritzsche received his PhD from the University of Göttingen in 1975, under the direction of Professor Hans Grauert. Since 1984, he has been Professor of Mathematics at the University of Wuppertal, where he has been investigating convexity problems on complex spaces and teaching undergraduate and graduate courses on Real and Complex Analysis. Hans Grauert studied physics and mathematics in Mainz, Münster and Zürich. He received his PhD in mathematics from the University of Münster and in 1959 he became a full professor at the University of Göttingen. Professor Grauert is responsible for many important developments in mathematics in the Twentieth Century. Along with Reinhold Remmert, Karl Stein and Henri Cartan, he founded the theory of Several Complex Variables in its modern form. He also proved various important theorems, including Levi's Problem and the coherence of higher direct image sheaves. Professor Grauert is the author of 10 books and his Selected Papers was published by Springer in 1994.
I 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.- Mer
* Holomorphic Functions * Domains of Holomorphy * Analytic Sets * Complex Manifolds * Stein Theory * Kaehler Manifolds * Boundary Behavior
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.