From Holomorphic Functions to Complex Manifolds

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.- Meromorphie Functions.- Grassmannian Manifolds.- Submanifolds and Normal Bundles.- Projective Algebraic Manifolds.- Projective Hypersurfaces.- The Euler Sequence.- Rational Functions.- 6. Branched Riemann Domains.- Branched Analytic Coverings.- Branched Domains.- Torsion Points.- Concrete Riemann Surfaces.- Hyperelliptic Riemann Surfaces.- 7. Modifications and Toric Closures.- Proper Modifications.- Blowing Up.- The Tautological Bundle.- Quadratic Transformations.- Monoidal Transformations.- Meromorphic Maps.- Toric Closures.- V Stein Theory.- 1. Stein Manifolds.- Fundamental Theorems.- Cousin-I Distributions.- Cousin-II Distributions.- Chern Class and Exponential Sequence.- Extension from Submanifolds.- Unbranched Domains of Holomorphy.- The Embedding Theorem.- The Serre Problem.- 2. The Levi Form.- Covariant Tangent Vectors.- Hermitian Forms.- Coordinate Transformations.- Plurisubharmonic Functions.- The Maximum Principle.- 3. Pseudoconvexity.- Pseudoconvex Complex Manifolds.- Examples.- Analytic Tangents.- 4. Cuboids.- Distinguished Cuboids.- Vanishing of Cohomology.- Vanishing on the Embedded Manifolds.- Cuboids in a Complex Manifold.- Enlarging U?.- Approximation.- 5. Special Coverings.- Cuboid Coverings.- The Bubble Method.- Fréchet Spaces.- Finiteness of Cohomology.- Holomorphic Convexity.- Negative Line Bundles.- Bundles over Stein Manifolds.- 6. The Levi Problem.- Enlarging: The Idea of the Proof.- Enlarging: The First Step.- Enlarging: The Whole Process.- Solution of the Levi Problem.- The Compact Case.- VI Kahler Manifolds.- 1. Differential Forms.- The Exterior Algebra.- Forms of Type (p, q).- Bundles of Differential Forms.- 2. Dolbeault Theory.- Integration of Differential Forms.- The Inhomogeneous Cauchy Formula.- The ??-Equation in One Variable.- A Theorem of Hartogs.- Dolbeault’s Lemma.- Dolbeault Groups.- 3. Kähler Metrics.- Hermitian metrics.- The Fundamental Form.- Geodesic Coordinates.- Local Potentials.- Pluriharmonic Functions.- The Fubini Metric.- Deformations.- 4. The Inner Product.- The Volume Element.- The Star Operator.- The Effect on (p, q)-Forms.- The Global Inner Product.- Currents.- 5. Hodge Decomposition.- Adjoint Operators.- The Kählerian Case.- Bracket Relations.- The Laplacian.- Harmonic Forms.- Consequences.- 6. Hodge Manifolds.- Negative Line Bundles.- Special Holomorphic Cross Sections.- Projective Embeddings.- Hodge Metrics.- 7. Applications.- Period Relations.- The Siegel Upper Halfplane.- Semipositive Line Bundles.- Moishezon Manifolds.- VII Boundary Behavior.- 1. Strongly Pseudoconvex Manifolds.- The Hilbert Space.- Operators.- Boundary Conditions.- 2. Subelliptic Estimates.- Sobolev Spaces.- The Neumann Operator.- Real-Analytic Boundaries.- Examples.- 3. Nebenhüllen.- General Domains.- A Domain with Nontrivial Nebenhülle.- Bounded Domains.- Domains in C2.- 4. Boundary Behavior of Biholomorphic Maps.- The One-Dimensional Case.- The Theory of Henkin and Vormoor.- Real-Analytic Boundaries.- Fefferman’s Result.- Mappings.- The Bergman Metric.- References.- Index of Notation.