Basic Algebraic Geometry 2

Table of Contents Volume 2.- BOOK 2. Schemes and Varieties.- V. Schemes.- 1. The Spec of a Ring.- 1.1. Definition of Spec A.- 1.2. Properties of Points of Spec A.- 1.3. The Zariski Topology of Spec A.- 1.4. Irreducibility, Dimension.- Exercises to §1.- 2. Sheaves.- 2.1. Presheaves.- 2.2. The Structure Presheaf.- 2.3. Sheaves.- 2.4. Stalks of a Sheaf.- Exercises to §2.- 3. Schemes.- 3.1. Definition of a Scheme.- 3.2. Glueing Schemes.- 3.3. Closed Subschemes.- 3.4. Reduced Schemes and Nilpotents.- 3.5. Finiteness Conditions.- Exercises to §3.- 4. Products of Schemes.- 4.1. Definition of Product.- 4.2. Group Schemes.- 4.3. Separatedness.- Exercises to §4.- VI. Varieties.- 1. Definitions and Examples.- 1.1. Definitions.- 1.2. Vector Bundles.- 1.3. Vector Bundles and Sheaves.- 1.4. Divisors and Line Bundles.- Exercises to §1.- 2. Abstract and Quasiprojective Varieties.- 2.1. Chow's Lemma.- 2.2. Blowup Along a Subvariety.- 2.3. Example of Non-Quasiprojective Variety.- 2.4. Criterions for Projectivity.- Exercises to §2.- 3. Coherent Sheaves.- 3.1. Sheaves of Ox-modules.- 3.2. Coherent Sheaves.- 3.3. Dévissage of Coherent Sheaves.- 3.4. The Finiteness Theorem.- Exercises to §3.- 4. Classification of Geometric Objects and Universal Schemes.- 4.1. Schemes and Functors.- 4.2. The Hilbert Polynomial.- 4.3. Flat Families.- 4.4. The Hilbert Scheme.- Exercises to §4.- Book 3. Complex Algebraic Varieties and Complex Manifolds.- VII. The Topology of Algebraic Varieties.- 1. The Complex Topology.- 1.1. Definitions.- 1.2. Algebraic Varieties as Differentiate Manifolds; Orientation.- 1.3. Homology of Nonsingular Projective Varieties.- Exercises to §1.- 2. Connectedness.- 2.1. Preliminary Lemmas.- 2.2. The First Proof of the Main Theorem.- 2.3. The Second Proof.- 2.4. Analytic Lemmas.- 2.5. Connectedness of Fibres.- Exercises to §2.- 3. The Topology of Algebraic Curves.- 3.1. Local Structure of Morphisms.- 3.2. Triangulation of Curves.- 3.3. Topological Classification of Curves.- 3.4. Combinatorial Classification of Surfaces.- 3.5. The Topology of Singularities of Plane Curves.- Exercises to §3.- 4. Real Algebraic Curves.- 4.1. Complex Conjugation.- 4.2. Proof of Harnack's Theorem.- 4.3. Ovals of Real Curves.- Exercises to §4.- VIII. Complex Manifolds.- 1. Definitions and Examples.- 1.1. Definition.- 1.2. Quotient Spaces.- 1.3. Commutative Algebraic Groups as Quotient Spaces.- 1.4. Examples of Compact Complex Manifolds not Isomorphic to Algebraic Varieties.- 1.5. Complex Spaces.- Exercises to §1.- 2. Divisors and Meromorphic Functions.- 2.1. Divisors.- 2.2. Meromorphic Functions.- 2.3. The Structure of the Field M(X).- Exercises to §2.- 3. Algebraic Varieties and Complex Manifolds.- 3.1. Comparison Theorems.- 3.2. Example of Nonisomorphic Algebraic Varieties that Are Isomorphic as Complex Manifolds.- 3.3. Example of a Nonalgebraic Compact Complex Manifold with Maximal Number of Independent Meromorphic Functions.- 3.4. The Classification of Compact Complex Surfaces.- Exercises to §3.- 4. Kahler Manifolds.- 4.1. Kähler Metric.- 4.2. Examples.- 4.3. Other Characterisations of Kähler Metrics.- 4.4. Applications of Kähler Metrics.- 4.5. Hodge Theory.- Exercises to §4tc].- IX. Uniformisation.- 1. The Universal Cover.- 1.1. The Universal Cover of a Complex Manifold.- 1.2. Universal Covers of Algebraic Curves.- 1.3. Projective Embedding of Quotient Spaces.- Exercises to §1.- 2. Curves of Parabolic Type.- 2.1. Theta functions.- 2.2. Projective Embedding.- 2.3. Elliptic Functions, Elliptic Curves and Elliptic Integrals.- Exercises to §2.- 3. Curves of Hyperbolic Type.- 3.1. Poincaré Series.- 3.2. Projective Embedding.- 3.3. Algebraic Curves and Automorphic Functions.- Exercises to §3.- 4. Uniformising Higher Dimensional Varieties.- 4.1. Complete Intersections are Simply Connected.- 4.2. Example of Manifold with ? a Given Finite Group.- 4.3. Remarks.- Exercises to §4.- Historical Sketch.- 1. Elliptic Integrals.- 2. Elliptic Functions.- 3. Abelian Integrals.- 4. Riemann Surfaces.- 5. The Inversion of Abelian Integrals.- 6. The Geometry of Algebraic Curves.- 7. Higher Dimensional Geometry.- 8. The Analytic Theory of Complex Manifolds.- 9. Algebraic Varieties over Arbitrary Fields and Schemes.- References.- References for the Historical Sketch.