Coherent Analytic Sheaves

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 Decomposition Theorem — 3. Global and Local Decomposition. Global Maximum Principle — 4. Proper Maps — 5. Holomorphically Spreadable Spaces.- § 3. Local and Arcwise Connectedness of Complex Spaces.- 1. Local Connectedness — 2. Arcwise Connectedness — 3. Finite Holomorphic Surjections and Covering Maps.- § 4. Removable Singularities of Analytic Sets.- 1. Analyticity of Closures of Coverings — 2. Extension Theorem for Analytic Sets — 3. Proof of Proposition 2–4. Historical Note.- § 5. Theorems of Chow, Levi and Hurwitz-Weierstrass.- 1. Theorem of Chow — 2. Levi Extension Theorem — 3. Theorem of Hurwitz-Weierstraß — 4. Historical Notes.- 10. Direct Image Theorem.- § 1. Polydisc Modules.- 1. The Protonorm System on O(E) — 2. Polydisc Modules — 3. Morphisms and Morphism Systems — 4. Complexes of Polydisc Modules — 5. Cohomology of Poly-disc Modules. Quasi-Isomorphisms — 6. Finiteness Lemma F(q) and Projection Lemma Z(q) for Cocycles.- § 2. Proof of Lemmata F(q) and Z(q).- 1. Homotopy — 2. Z(q) ? Z(q-1) — 3. F(q), Z(q)?F(q-1) begin — 4. Smoothing — 5. Construction of Lq-1, ? - 6. Basic Property of ? - 7. Vanishing of Hq-1(t, ?, K).- § 3. Sheaves of Polydisc Modules.- 1. Definitions for $$U \subset \dot E$$ — 2. The Natural Functor — 3. The Paragraphs 1.4–1.6 for Polydisc Sheaves — 4. Coherence of Cohomology Sheaves. Main Theorem.- § 4. Coherence of Direct Image Sheaves.- 1. Mounting Complex Spaces — 2. Resolutions — 3. Complexes of Polydisc Modules — 4. Complexes of Sheaves — 5. Application of the Main Theorem — 6. The Direct Image Theorem.- § 5. Regular Families of Compact Complex Manifolds.- 1. Regular Families — 2. Complex Subspaces Y’ ? Y of Codimension 1 — 3. The Maps fy,i — 4. Upper Semi-Continuity — 5. The Case $${\dim _C}{H^i}\left( {{X_y},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V} }_y}} \right) = $$ constant — 6. Rigid Complex Manifolds.- § 6. Stein Factorization and Applications.- 1. Stein Factorization of Proper Holomorphic Maps — 2. Proper Modifications of Normal Complex Spaces — 3. Graph of a Finite System of Meromorphic Functions — 4. Analytic and Algebraic Dependence — 5. Base Space of a Finite System of Meromorphic Functions — 6. Properties of Base Spaces — 7. Analytic Closures and Structure of the Field M(X) — 8. Reduction Theorem for Holomorphically Convex Spaces.- Annex. Theory of Sheaves. Notion of Coherence.- §0. Sheaves.- 1. Sheaves and Morphisms — 2. Restrictions, Subsheaves and Sums of Sheaves — 3. Sections. Hausdorff Sheaves.- § 1. Construction of Sheaves from Presheaves.- 1. Presheaves — 2. The Sheaf Associated to a Preshaf — 3. Canonical Presheaves — 4. Image Sheaves.- § 2. Sheaves and Presheaves with Algebraic Structure.- 1. Sheaves of Groups, Rings and A-Modules — 2. The Category of A-Modules. Quotient Sheaves — 3. Presheaves with Algebraic Structure — 4. The Functor Hom — 5. The Functor ?.- § 3. Coherent Sheaves.- 1. Sheaves of Finite Type — 2. Sheaves of Relation Finite Type — 3. Coherent Sheaves.- § 4. Yoga of Coherent Sheaves.- 1. Three Lemma — 2. Consequences of the Three Lemma — 3. Coherence of Trivial Extensions — 4. Coherence of the Functors Hom and ? — 5. Annihilator Sheaves.- Index of Names.