Inhalt

1. Complex Spaces.- § 1. The Notion of a Complex Space.- 0. Ringed Spaces - 1. The Space (?n, (O) - 2. Zero Sets and Complex Model Spaces - 3. Sheaves of Local ?-Algebras. ?-ringed Spaces - 4. Morphisms of ?-ringed Spaces - 5. Complex Spaces - 6. Sections and Functions - 7. Construction of Complex Spaces by Gluing - 8. The Complex Projective Space ?n - 9. Historical Notes.- § 2. General Properties of Complex Spaces.- 1. Zero Sets of Ideal Sheaves - 2. Closed Complex Subspaces - 3. Factorization of Holomorphic Maps - 4. Complex Spaces and Coherent Analytic Sheaves. Extension Principle - 5. Analytic Image Sheaves - 6. Analytic Inverse Image Sheaves - 7. Holomorphic Embeddings.- § 3. Direct Products and Graphs.- 1. The Bijection ?ol(X, ?n)?O(X)n. Extension of Holomorphic Maps - 2. Complex Direct Products - 3. Existence of Canonical Products. Local Case - 4. Existence of Canonical Products. Global Case - 5. Graph Space of a Holomorphic Map.- § 4. Complex Spaces and Cohomology.- 1. Divisors - 2. Holomorphic Vector Bundles - 3. Line Bundles and Divisors - 4. Holomorphically Convex Spaces and Stein Spaces - 5. ?ech Cohomology of Analytic Sheaves - 6. Cohomology of Coherent Sheaves with Respect to Stein Coverings - 7. Higher Dimensional Direct Images.- 2. Local Weierstrass Theory.- § 1. The Weierstrass Theorems.- 0. Generalities - 1. The WeierstraB Division Theorem - 2. The Weierstraß Preparation Theorem - 3. A Simple Observation.- § 2. Algebraic Structure of $${O_{{C^n},0}}$$.- 1. Noether Property and Factoriality - 2. Hensel's Lemma - 3. Closedness of Sub-modules.- § 3. Finite Maps.- 1. Closed Maps - 2. Finite Maps. Local Description - 3. Local Representation of Image Sheaves - 4. Exactness of the Functor f* for Finite Maps - 5. Weierstraß Maps.- §4. The Weierstrass Isomorphism.- 1. The Generalized Weierstraß Division Theorem - 2. The Weierstraß Isomorphism - 3. A Coherence Lemma - 4. A Further Generalization of the Generalized Weierstraß Division Theorem.- § 5. Coherence of Structure Sheaves.- 1. Formal Coherence Criterion - 2. The Coherence of $${O_{{C^n}}}$$ - 3. Coherence of all Structure Sheaves OX.- 3. Finite Holomorphic Maps.- § 1. Finite Mapping Theorem.- 1. Projection Lemma - 2. Finite Holomorphic Maps and Isolated Points - 3. Finite Mapping Theorem.- § 2. Rückert Nullstellensatz for Coherent Sheaves.- 1. Preliminary Version - 2. Rückert Nullstellensatz.- § 3. Finite Open Holomorphic Maps.- 1. A Necessary Condition for Openness - 2. Torsion Sheaves and Criterion of Openness - 3. Coherence of Torsion Sheaves and Open Mapping Lemma - 4. Existence of Finite Open Projections.- § 4. Local Description of Complex Subspaces in ?n.- 1. The Local Description Lemma - 2. Proof of the Local Description Lemma.- 4. Analytic Sets. Coherence of Ideal Sheaves.- § 1. Analytic Sets and their Ideal Sheaves.- 1. Analytic Sets - 2. Ideal Sheaf of an Analytic Set - 3. Local Decomposition Lemma - 4. Prime Components. Criterion of Reducibility - 5. Rückert Nullstellensatz for Ideal Sheaves - 6. Analytic Sets and Finite Holomorphic Maps.- § 2. Coherence of the Sheaves i (A).- 1. Proof of Coherence in a Special Case - 2. Reduction to Analytic Sets in Domains of ?n - 3. Further Reduction to a Lemma - 4. Verification of the Assumptions of Lemma 3-5. Coherence of Radical Sheaves.- § 3. Applications of the Fundamental Theorem and of the Nullstellensatz.- 1. Analytic Sets and Reduced Closed Complex Subspaces - 2. Reduction of Complex Spaces - 3. Reduced Complex Spaces.- § 4. Coherent and Locally Free Sheaves.- 1. Corank of a Coherent Sheaf - 2. Characterization of Locally Free Sheaves.- 5. Dimension Theory.- § 1. Analytic and Algebraic Dimension.- 1. Analytic Dimension of Complex Spaces. Upper Semi-Continuity - 2. Analytic and Algebraic Dimension - 3. Dimension of the Reduction and of Analytic Sets.- § 2. Active Germs and the Active Lemma.- 1. The Sheaf of Active Germs - 2. Criterion of Activity - 3. Existence of Active Functions. Lifting Lemma - 4. Active Lemma.- § 3. Applications of the Active Lemma.- 1. Basic Properties of Dimension. Ritt's Lemma - 2. Analytic Sets of Maximal Dimension - 3. Computation of the Dimension of Analytic Sets in ?n.- § 4. Dimension and Finite Maps. Pure Dimensional Spaces.- 1. Invariance of Dimension under Finite Maps - 2. Pure Dimensional Complex Spaces - 3. Open Finite Maps and Dimension. Open Mapping Theorem - 4. Local Prime Components (revisited).- § 5. Maximum Principle.- 1. Open Mapping Theorem for Holomorphic Functions - 2. Local and Absolute Maximum Principle - 3. Maximum Principle for Complex Spaces with Boundary.- § 6. Noether Lemma for Coherent Analytic Sheaves.- 1. Statement of the Lemma and Applications - 2. Proof of the Lemma.- 6. Analyticity of the Singular Locus. Normalization of the Structure Sheaf.- § 1. Embedding Dimension.- 1. Embedding Dimension. Jacobi Criterion - 2. Analyticity of the Sets X(k). Algebraic Description of embxX.- § 2. Smooth Points and the Singular Locus.- 1. Smooth Points and Singular Locus - 2. Analyticity of the Singular Locus - 3. A Property of the Ideals i(S(X))x, x?S(X).- § 3. The Sheaf M of Germs of Meromorphic Functions.- 1. The Sheaf M - 2. The Zero Set and the Polar Set of a Meromorphic Function - 3. The Lifting Monomorphism MY?f*(MX).- § 4. The Normalization Sheaf $${\hat O_X}$$.- 1. The Normalization Sheaf Normal Points $${\hat O_X}$$ - 2. Normality and Irreducibility at a Point.- § 5. Criterion of Normality. Theorem of Oka.- 1. The Canonical OX homomorphism $$\sigma :Hom\left( {f,f} \right) \to M$$ - 2. Criterion of Normality. Theorem of Oka - 3. Singular Locus and Normal Points.- 7. Riemann Extension Theorem and Analytic Coverings.- § 1. Riemann Extension Theorem on Complex Manifolds.- 1. First Riemann Theorem - 2. Second Riemann Theorem - 3. Riemann Extension Theorem on Complex Manifolds. Criterion of Connectedness.- § 2. Analytic Coverings.- 1. Definition and Elementary Properties - 2. Covering Lemma and Existence of Open Coverings - 3. Open Analytic Coverings.- § 3. Theorem of Primitive Element.- 1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism $${\pi }\left( {{{\hat O}X}} \right) \to O_Y^b$$.- § 4. Applications of the Theorem of Primitive Element.- 1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces - 2. Characterization of Normality by the Riemann Extension Theorem - 3. Weierstraß Convergence Theorem on Locally Pure Dimensional Complex Spaces.- § 5. Analytically Normal Vector Bundles.- 1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles - 5. Discussion of the Cones Akm.- 8. Normalization of Complex Spaces.- § 1. One-Sheeted Analytic Coverings.- 1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms $$\tilde v:{M_Y}\tilde \to {\tilde v}\left( {{M_X}} \right) $$ and $$\tilde v:{\hat O_Y}\tilde \to {v*}\left( {{{\hat O}_X}} \right)$$.- § 2. The Local Existence Theorem. Coherence of the Normalization Sheaf.- 1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf.- § 3. The Global Existence Theorem. Existence of Normalization Spaces.- 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization.- § 4. Properties of the Normalization.- 1. The Space of Prime Germs. Topological Structure of Normalization Spaces - 2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps.- 9. Irreducibility and Connectivity. Extension of Analytic Sets.- § 1. Irreducible Complex Spaces.- 1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces.- § 2. Global Decomposition of Complex Spaces.- 1. Connected Components - 2. Global Decomposi…