Book 2： Schemes and Varieties
5 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
1.5 Exercises to Section 1
2 Sheaves
2.1 Presheaves
2.2 The Structure Presheaf
2.3 Sheaves
2.4 Stalks of a Sheaf
2.5 Exercises to Section 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
3.6 Exercises to Section 3
4 Products of Schemes
4.1 Definition of Product
4.2 Group Schemes
4.3 Separatedness
4.4 Exercises to Section 4
6 Variehes
1 Definitions and Examples
1.1 Definitions
1.2 Vector Bundles
1.3 Vector Bundles and Sheaves
1.4 Divisors and Line Bundles
1.5 Exercises to Section 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
2.5 Exercises to Section 2
3 Coherent Sheaves
3.1 Sheaves of Ox—Modules
3.2 Coherent Sheaves
3.3 Devissage of Coherent Sheaves
3.4 The Finiteness Theorem
3.5 Exercises to Section 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
4.5 Exercises to Section 4
Book 3： Complex Algebraic Varieties and Complex Manifolds
7 The Topology of Algebraic Varieties
1 The Complex Topology
1.1 Definitions
1.2 Algebraic Varieties as Differentiable Manifolds；Orientation
1.3 Homology of Nonsingular Projective Varieties
1.4 Exercises to Section 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
2.6 Exercises to Section 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
3.6 Exercises to Section 3
4 Real Algebraic Curves
4.1 Complex Conjugation
4.2 Proof of Harnack's Theorem
4.3 Ovals of Real Curves
4.4 Exercises to Section 4
8 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
1.6 Exercises to Section 1
2 Divisors and Meromorphic Functions
2.1 Divisors
2.2 Meromorphic Functions
2.3 The Structure of the Field M（X）
2.4 Exercises to Section 2
3 Algebraic Varieties and Complex Manifolds
3.1 Comparison Theorems
3.2 Example of Nonisomorphic Algebraic Varieties that Are Isomorphic as Complex Mani
3.3 Example of a Nonalgebraoc Compact Complex Manifold with Maximal Number of Independent Meromorphic Functions
3.4 The Classification of Compact Complex Surfaces
3.5 Exercises to Section 3
4 Kahler Manifolds
4.1 Kahler Metric
4.2 Examples
4.3 Other Characterisations of Kahler Metrics
4.4 Applications of Kahler Metrics
4.5 Hodge Theory
4.6 Exercises to Section 4
9 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
1.4 Exercises to Section 1
2 Curves of Parabolic Type
2.1 Theta Functions
2.2 Projective Embedding
2.3 Elliptic Functions，Elliptic Curves and Elliptic Integrals
2.4 Exercises to Section 2
3 Curves of Hyperbolic Type
3.1 Poincare Series
3.2 Projective Embedding
3.3 Algebraic Curves and Automorphic Functions
3.4 Exercises to Section 3
4 Uniformising Higher Dimensional Varieties
4.1 Complete Intersections are Simply Connected
4.2 Example of Manifold with π1 a Given Finite Group
4.3 Remarks
4.4 Exercises to Section 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
Index
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Book 1:Varieties in projective space
