目錄
Preface
Contents
Introduction
1 categories,Froducts,Projective and Inductive Limits
1.1 The Notion of a Category and Examples
1.2 Functors
1.3 Products,Projective Limits and Direct Limits in a Category
1.3.1 The Projective Limit
1.3.2 The Yoneda Lemma
1.3.3 Examples
1.3.4 Representable Functors
1.3.5 Direct Limits
1.4 Exercises
2 Basic Concepts of Homological AlgSbra
2.1 The Category Modr of r-modules
2.2 More Functors
2.2.1 Invariants,Coinvariants and Bxactnes
2.2.2 The First Cohomology Group
2.2.3 Some Notation
2.2.4 Exercises
2.3 The Derived Functors
2.3.1 The Simple Principle
2.3.2 Functoriality
2.3.3 Other Resolutions
2.3.4 Injective Resolutions of Short Exact Sequences
A Fundamental Remark
The Cohomology and the Long Bxact Sequence
The Homology of Groups
2.4 The Functors Ext and Tor
2.4.1 The Functor Ext
2.4.2 The Derived Functor for the Tensor Product
2.4.3 Exercise
3 Sheaves
3.1 Presheaves and Sheaves
3.1.1 What is a Presheaf
3.1.2 A Remark about Products and Presheaf
3.1.3 What is a Sheaf
3.1.4 Examples
3.2 Manifolds as Locally Ringed Spaces
3.2.1 What Are Manifolds
3.2.2 Examples and Exercise
3.3 Stalks and Sheafification
3.3.1 Stalks
3.3.2 The Process of Sheafification of a Presheaf
3.4 The Functors f*and f
3.4.1 The Adjunction Formula
3.4.2 Bxtensions and Restrictiona
3.5 Constructions of Sheaves
4 Cohomology of Sheaves
4.1 Examples
4.1.1 .Sheaves on Riemann surfaces
4.1.2 Cohomology of the Circle
4.2 The Derived Functor
4.2.1 Injective Sheaves and Derived Functors
4.2.2 A Direct Definition of H1
4.3 Fiber Bundles and Non Abelian H1
4.3.1 Fibrations
Fibre Bundle
Vector Bundles
4.3.2 Non-Abelian H1
4.3.3 The Reduction of the Structure Group
Orientation
Local Systems
Isomorphism Classes of Local Systems
Principal G-bundels
4.4 Fundamental Properties of the Cohomology of Sheaves
4.4.1 Introduction
4.4.2 The Derived Functor to f
4.4.3 Functorial Properties of the Cohomology
4.4.4 Paracompact Spaces
4.4.5 Applications
Cohomology of Spheres
Orientations
Compact Oriented Surfaces
4.5 Cech Cohomology of Sheaves
4.5.1 The Cech-Complex
4.5.2 The Cech Resolution of a Sheaf
4.6 Spectral Sequences
4.6.1 Introduction
4.6.2 The Vertical Filtration
4.6.3 The Horizontal Filtration
Two Special Cases
Applications of Spectral Sequences
4.6.4 The Derived Category
The Composition Rule
Exact Triangles
4.6.5 The Spectral Sequence of a Fibration
Sphere Bundles an Euler Characteristic
4.6.6 Cech Complexes and the Spectral Sequence
A Criterion for Degeneration
An Application to Product Spaces
4.6.7 The Cup Product
4.6.8 Example: Cup Product for the Comology of Tori
A Connection to the Cohomology of Groups
4.6.9 An Excursion into Homotopy Theory
4.7 Cohomology with Compact Supports
4.7.1 The Definition
4.7.2 An Example for Cohomology with Compact Supports
The Cohomology with Compact Supports for Open Balls
Formulae for Cup Products
4.7.3 The Fundamental Class
4.8 Cohomology of Manifolds
4.8.1 Local Systems
4.8.2 Cech Resolutions of Local Systems
4.8.3 Cech Coresolution of Local Systems
4.8.4 Poincare Duality
4.8.5 The Cohomology in Top Degree and the Homology
4.8.6 Some Remarks on Singular Homology
4.8.7 Cohomology with Compact Support and Embeddings
4.8.8 The Fundamental Class of a Submanifold
4.8.9 Cup Product and Intersections
4.8.10 Compact oriented Surfaces
4.8.11 The Cohomology Ring of Pn (C)
4.9 The Lefschetz Fixed Point Formula
4.9.1 The Euler Characteristic of Manifolds
4.10 The de Rham and the Dolbeault Isomorphism
4.10.1 The Cohomology of Flat Bundles on Real Manifolds
The Product Structure on the de Rham Cohomology
The de Rham Isomorphism and the fundamental class
4.10.2 Cohomology of Holomorphic Bundles on Complex Manifolds
The Tangent Bundle
The Bundle SRg
4.10.3 Chern Classes
The Line Bundles Opn (c)(k)
4.11 Hodge Theory
4.11.1 Hodge Theory on Real Manifolds
4.11.2 Hodge Theory on Complex Manifolds
Some Linear Algebra
Kahler Manifolds and their Cohomology
The Cohomology of Holomorphic Vector Bundles
Serre Duality
4.11.3 Hodge Theory on Tori
5 Compact Riemann surfaces and Abelian Varieties
5.1 Compact Riemann Surfaces
5.1.1 Introduction
5.1.2 The Hodge Structure on H1(S,C)
5.1.3 Cohomology of Holomorphic Bundles
5.1.4 The Theorem of Riemann-Roch
On the Picard Group
Exercises
The Theorem of Riemann-Roch
5.1.5 The Algebraic Duality Pairing
5.1.6 Riemann Surfaces of Low Genus
5.1.7 The Algebraicity of Riemann Surfaces
From a Riemann Surface to Function Fields
The reconstruction of S from K
Connection to Algebraic Geometry
Elliptic Curves
5.1.8 Geometrie Analytique et Geometrie Algebrique-GAGA
5.1.9 Comparison of Two Pairings
5.1.10 The Jacobian of a Compact Riemann Surface
5.1.11 The Classical Version of Abel's Theorem
5.1.12 Riemann Period Relations
5.2 Line Bundles on Complex Tori
5.2.1 Construction of Line Bundles
The Poincare Bundle
Universality of N
5.2.2 Homomorphisms Between Complex Tori
The Neron Severi group and Hom (A,AV)
The construction of y starting from a line bundle
5.2.3 The Self Duality of the Jacobian
5.2.4 Ample Line Bundles and the Algebraicity of the Jacobian
The Kodaira Embedding Theorem
The Spaces of Sections
5.2.5 The Siegel Upper Half Space
Elliptic curves with level structure
The end of the excursion
5.2.6 Riemann-Theta Functions
5.2.7 Projective embeddings of abelian varieties
5.2.8 Degeneration of Abelian Varieties
The Case of Genus 1
The Algebraic Approach
5.3 Towards the Algebraic Theory
5.3.1 Introduction
The Algebraic Definition of the Neron-Severi Group
The Algebraic Definition of the Intersection Numbers
The Study of some Special Neron-Severi groups
5.3.2 The Structure of End (J)
The Rosati Involution
A Trace Formula
The Fundamental Class [S] of S under the Abel Map
5.3.3 The Ring of Correspondences
5.3.4 An Algebraic Substitute for the Cohomology
Bibliography
Index
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