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Preface, with an Extended Introduction; Part I Background Material; 1 General Set-Up; 1.1 Varieties; 1.1.1 Linearising the Category of Varieties; 1.1.2 Divisors with Normal Crossings; 1.2 Complex Analytic Spaces; 1.2.1 Analytification; 1.3 Complexes; 1.3.1 Basic Definitions; 1.3.2 Filtrations; 1.3.3 Total Complexes and Signs; 1.4 Hypercohomology; 1.4.1 Definition; 1.4.2 Godement Resolutions; 1.4.3 Čech Cohomology; 1.5 Simplicial Objects; 1.6 Grothendieck Topologies; 1.7 Torsors; 1.7.1 Sheaf-Theoretic Definition; 1.7.2 Torsors in the Category of Sets
1.7.3 Torsors in the Category of Schemes (Without Groups)2 Singular Cohomology; 2.1 Relative Cohomology; 2.2 Singular (Co)homology; 2.3 Simplicial Cohomology; 2.4 The Künneth Formula and Poincaré Duality; 2.5 The Basic Lemma; 2.5.1 Formulations of the Basic Lemma; 2.5.2 Direct Proof of Basic Lemma I; 2.5.3 Nori's Proof of Basic Lemma II; 2.5.4 Beilinson's Proof of Basic Lemma II; 2.5.5 Perverse Sheaves and Artin Vanishing; 2.6 Triangulation of Algebraic Varieties; 2.6.1 Semi-algebraic Sets; 2.6.2 Semi-algebraic Singular Chains; 2.7 Singular Cohomology via the h'-Topology
3 Algebraic de Rham Cohomology3.1 The Smooth Case; 3.1.1 Definition; 3.1.2 Functoriality; 3.1.3 Cup Product; 3.1.4 Change of Base Field; 3.1.5 Étale Topology; 3.1.6 Differentials with Log Poles; 3.2 The General Case: Via the h-Topology; 3.3 The General Case: Alternative Approaches; 3.3.1 Deligne's Method; 3.3.2 Hartshorne's Method; 3.3.3 Using Geometric Motives; 3.3.4 The Case of Divisors with Normal Crossings; 4 Holomorphic de Rham Cohomology; 4.1 Holomorphic de Rham Cohomology; 4.1.1 Definition; 4.1.2 Holomorphic Differentials with Log Poles; 4.1.3 GAGA
4.2 Holomorphic de Rham Cohomology via the h'-Topology4.2.1 h'-Differentials; 4.2.2 Holomorphic de Rham Cohomology; 4.2.3 GAGA; 5 The Period Isomorphism; 5.1 The Category (k,mathbbQ)-Vect; 5.2 A Triangulated Category; 5.3 The Period Isomorphism in the Smooth Case; 5.4 The General Case (via the h'-Topology); 5.5 The General Case (Deligne's Method); 6 Categories of (Mixed) Motives; 6.1 Pure Motives; 6.2 Geometric Motives; 6.3 Absolute Hodge Motives; 6.4 Mixed Tate Motives; Part II Nori Motives; 7 Nori's Diagram Category; 7.1 Main Results; 7.1.1 Diagrams and Representations
7.1.2 Explicit Construction of the Diagram Category7.1.3 Universal Property: Statement; 7.1.4 Discussion of the Tannakian Case; 7.2 First Properties of the Diagram Category; 7.3 The Diagram Category of an Abelian Category; 7.3.1 A Calculus of Tensors; 7.3.2 Construction of the Equivalence; 7.3.3 Examples and Applications; 7.4 Universal Property of the Diagram Category; 7.5 The Diagram Category as a Category of Comodules; 7.5.1 Preliminary Discussion; 7.5.2 Coalgebras and Comodules; 8 More on Diagrams; 8.1 Multiplicative Structure; 8.2 Localisation; 8.3 Nori's Rigidity Criterion
1.7.3 Torsors in the Category of Schemes (Without Groups)2 Singular Cohomology; 2.1 Relative Cohomology; 2.2 Singular (Co)homology; 2.3 Simplicial Cohomology; 2.4 The Künneth Formula and Poincaré Duality; 2.5 The Basic Lemma; 2.5.1 Formulations of the Basic Lemma; 2.5.2 Direct Proof of Basic Lemma I; 2.5.3 Nori's Proof of Basic Lemma II; 2.5.4 Beilinson's Proof of Basic Lemma II; 2.5.5 Perverse Sheaves and Artin Vanishing; 2.6 Triangulation of Algebraic Varieties; 2.6.1 Semi-algebraic Sets; 2.6.2 Semi-algebraic Singular Chains; 2.7 Singular Cohomology via the h'-Topology
3 Algebraic de Rham Cohomology3.1 The Smooth Case; 3.1.1 Definition; 3.1.2 Functoriality; 3.1.3 Cup Product; 3.1.4 Change of Base Field; 3.1.5 Étale Topology; 3.1.6 Differentials with Log Poles; 3.2 The General Case: Via the h-Topology; 3.3 The General Case: Alternative Approaches; 3.3.1 Deligne's Method; 3.3.2 Hartshorne's Method; 3.3.3 Using Geometric Motives; 3.3.4 The Case of Divisors with Normal Crossings; 4 Holomorphic de Rham Cohomology; 4.1 Holomorphic de Rham Cohomology; 4.1.1 Definition; 4.1.2 Holomorphic Differentials with Log Poles; 4.1.3 GAGA
4.2 Holomorphic de Rham Cohomology via the h'-Topology4.2.1 h'-Differentials; 4.2.2 Holomorphic de Rham Cohomology; 4.2.3 GAGA; 5 The Period Isomorphism; 5.1 The Category (k,mathbbQ)-Vect; 5.2 A Triangulated Category; 5.3 The Period Isomorphism in the Smooth Case; 5.4 The General Case (via the h'-Topology); 5.5 The General Case (Deligne's Method); 6 Categories of (Mixed) Motives; 6.1 Pure Motives; 6.2 Geometric Motives; 6.3 Absolute Hodge Motives; 6.4 Mixed Tate Motives; Part II Nori Motives; 7 Nori's Diagram Category; 7.1 Main Results; 7.1.1 Diagrams and Representations
7.1.2 Explicit Construction of the Diagram Category7.1.3 Universal Property: Statement; 7.1.4 Discussion of the Tannakian Case; 7.2 First Properties of the Diagram Category; 7.3 The Diagram Category of an Abelian Category; 7.3.1 A Calculus of Tensors; 7.3.2 Construction of the Equivalence; 7.3.3 Examples and Applications; 7.4 Universal Property of the Diagram Category; 7.5 The Diagram Category as a Category of Comodules; 7.5.1 Preliminary Discussion; 7.5.2 Coalgebras and Comodules; 8 More on Diagrams; 8.1 Multiplicative Structure; 8.2 Localisation; 8.3 Nori's Rigidity Criterion